The Mertens and Polya´ Conjectures in Function Fields

Peter Humphries

August 2012

A thesis submitted for the degree of Master of Philosophy of The Australian National University

Declaration

The work in this thesis is my own except where otherwise stated.

Peter Humphries

Acknowledgements

Many thanks go to my supervisor, Jim Borger, whose willingness to take me on board was a godsend. I must also thank Andrew Hassell for his guidance and patience with me. I’d also like to thank Richard Brent for first introducing me to the problems in this thesis, and Tim Trudgian and Byungchul Cha for conversations and advice on my work. Finally, I wish to thank my friends and family for all their support over the last eighteen months. I’m going to miss each and every one of you.

v

Abstract

The Mertens conjecture on the order of growth of the summatory function of the M¨obiusfunction has long been known to be false. We formulate an analogue of this conjecture in the setting of global function fields, and investigate the plausibility of this conjecture. First we give certain conditions, in terms of the zeroes of the associated zeta functions, for this conjecture to be true. We then show that in a certain family of function fields of low genus, the average proportion of curves satisfying the Mertens conjecture is zero, and we hypothesise that this is true for any genus. Finally, we also formulate a function field version of P´olya’s conjecture, and prove similar results.

vii

Contents

Acknowledgements v

Abstract vii

1 The Mertens Conjecture in Function Fields 1 1.1 The Mertens Conjecture ...... 1 1.2 Mertens Conjectures in Function Fields ...... 3

2 Local Mertens Conjectures 9

2.1 An Explicit Expression for MC/Fq (X)...... 9

2.2 MC/Fq (X) and Zeroes of Multiple Order ...... 16 X/2 2.3 The Limiting Distribution of MC/Fq (X)/q ...... 20

3 Examples in Low Genus 27 3.1 Elliptic Curves over Finite Fields ...... 27 3.2 Proof of Theorem 1.7 ...... 31

4 Global Mertens Conjectures 37 4.1 Averages over Families of Curves ...... 37 4.2 The Critical Point ...... 47 4.3 Variants of the Mertens Conjecture ...... 53

5 P´olya’s Conjecture in Function Fields 57 5.1 P´olya’s Conjecture ...... 57 5.2 P´olya Conjectures in Function Fields ...... 59

6 Local P´olya Conjectures 63

6.1 An Explicit Expression for LC/Fq (X) ...... 63 6.2 Examples in Low Genus ...... 70 X/2 6.3 The Limiting Distribution of LC/Fq (X)/q ...... 74

ix x Contents

7 Global P´olya Conjectures 79 7.1 Averages over Families of Curves ...... 79 7.2 Variants of P´olya’s Conjecture ...... 85

A Proof of the Kronecker–Weyl Theorem 89

Bibliography 95 Chapter 1

The Mertens Conjecture in Function Fields

1.1 The Mertens Conjecture

Let µ(n) denote the M¨obiusfunction, so that for a positive integer n,  1 if n = 1,  µ(n) = (−1)t if n is the product of t distinct primes,  0 if n is divisible by a perfect square.

The Mertens conjecture states that the summatory function of the M¨obiusfunc- tion, X M(x) = µ(n), n≤x satisfies the inequality √ |M(x)| ≤ x (1.1) for all x ≥ 1. This conjecture stems from the work of Mertens [17], who in 1897 calculated M(x) from x = 1 up to x = 10 000 and arrived at the conjecture (1.1). Notably, this conjecture implies that all of the nontrivial zeroes of the Riemann zeta function ζ(s) lie on the line <(s) = 1/2 (that is, that the Riemann hypothesis is true), and also that all such zeroes are simple. However, Ingham [12] later showed that the Mertens conjecture implies that the imaginary parts of the zeroes of ζ(s) in the upper half-plane must be linearly dependent over the rational numbers, a relation that seems unlikely; while there is yet to be found strong theoretical evidence for the falsity of such a linear dependence,

1 2 The Mertens Conjecture in Function Fields

some limited numerical calculations have failed to find any such linear relations [1]. Using methods closely related to the work of Ingham, Odlyzko and te Riele [22] disproved the Mertens conjecture, and in fact showed that

M(x) lim sup √ > 1.06, x→∞ x M(x) lim inf √ < −1.009. x→∞ x

These bounds have since been improved to 1.218 and −1.229 respectively [15]. Despite this disproof, a single counterexample to the Mertens conjecture has yet to be found. Indeed, numerical calculations of Amir Akbary and Nathan Ng (personal communication), based on the paper [21] of Ng, suggest that the set of counterexamples to the Mertens conjecture is sparsely distributed in [1, ∞). More precisely, under the assumption of several strong yet plausible conjectures, they have shown that the logarithmic density 1 Z dx δ (Pµ) = lim X→∞ log X x Pµ∩[1,X] √ of the set Pµ = {x ∈ [1, ∞): |M(x)| ≤ x} is extremely close to 1 but strictly less than 1, satisfying the bounds

0.99999927 < δ (Pµ) < 1. (1.2) √ So although the Mertens conjecture is false, the inequality |M(x)| ≤ x nev- ertheless seems to hold for “most” x ≥ 1; on the other hand, the set of x for which this inequality fails to hold is nevertheless nontrivial, in the sense that it has strictly positive, albeit extremely small, logarithmic density. The reason for this stems from the following explicit expression for M(x) in terms of a sum over the nontrivial zeroes ρ of the Riemann zeta function.

Proposition 1.1 (Ng [21]). Assume the Riemann hypothesis and the simplicity ∞ of the zeroes of ζ(s). Then there exists a sequence {Tv}v=1 with v ≤ Tv ≤ v + 1 such that for each positive integer v, for all ε > 0, and for x a positive noninteger,

X 1 xρ  x log x x  M(x) = 0 + Oε 1 + + 1−ε , ζ (ρ) ρ Tv Tv log x |γ|

|γ| < Tv. 1.2 Mertens Conjectures in Function Fields 3

So we see that for x a noninteger,

M(x) X 1 xiγ  1  √ = + O √ , (1.3) x ζ0(ρ) ρ x ρ where the sum P is interpreted in the sense lim P . Now the coefficients ρ v→∞ |γ|

1.2 Mertens Conjectures in Function Fields

A natural variant of this problem is to formulate a function field analogue of the Mertens conjecture and determine how often this conjecture holds. The advantage of this function field setting, as opposed to the classical case, is that we may prove unconditional results about the behaviour of the summatory function of the M¨obiusfunction function. In the function field setting, the Riemann hypothesis is proved, and the hypothesis that the imaginary parts of the zeroes of the zeta function of a function field are linearly independent over the rational numbers — that is to say, the Linear Independence hypothesis — is, at the very least, true in an averaged sense; see Theorem 4.4 for the precise formulation of this result. Throughout this thesis, the function fields we work with will be global function fields, that is, of transcendence degree one over a finite constant field; equivalently, these are the function field of a nonsingular projective curve over a finite field. We define the M¨obiusfunction of a function field as follows. Let q = pm be an odd prime power, and let Fq be a finite field with q elements. Let C be a nonsingular projective curve over Fq of genus g; we write C/Fq for the function field of C over Fq. Then for an effective divisor D of C, the M¨obiusfunction of 4 The Mertens Conjecture in Function Fields

C/Fq is given by  1 if D is the zero divisor,  t µC/Fq (D) = (−1) if D is the sum of t distinct prime divisors,  0 if a prime divisor divides D with order at least 2.

We are interested in the summatory function of the M¨obiusfunction of C/Fq:

X−1 X X MC/Fq (X) = µC/Fq (D), N=0 deg(D)=N where X is a positive integer. We wish to determine the validity of the following conjecture.

The Mertens Conjecture in Function Fields. Let C be a nonsingular pro- jective curve over Fq of genus g, and let MC/Fq (X) be the summatory function of the M¨obiusfunction of C/Fq. Then

MC/Fq (X) lim sup X/2 ≤ 1. X→∞ q √ The presence of qX/2 in the denominator, as opposed to x in the classical case, is due to the fact that we are summing over divisors D with deg(D) ≤ X −1, whose absolute norms N D are qdeg(D), as opposed to the classical case where we sum over all positive integers n ≤ x, whose norm in each case is simply n itself. Several natural questions arise from formulating this conjecture. We may first ask “local” questions: given a curve, how do we determine whether the Mertens conjecture for the function field of this curve holds?

Question 1.2. For which curves does the Mertens conjecture hold?

An further local problem is to consider the Mertens conjecture for each positive integer X.

Question 1.3. Given a function field C/Fq, how frequently does the inequality

X/2 MC/Fq (X) ≤ q (1.4) hold?

As there are many function fields of a given genus g over a finite field q, we may also consider a “global” question on the Mertens conjecture. 1.2 Mertens Conjectures in Function Fields 5

Question 1.4. On average, in either the q or the g aspect, how often does the Mertens conjecture hold?

We treat the local questions in Chapter 2. There we find that the major difference to the classical case is that Question 1.2 is non-trivial: there do exist curves for which the Mertens conjecture is true. In Section 2.1, we formulate certain conditions on the zeroes of ZC/Fq (u), the zeta function of C/Fq, to ensure that the Mertens conjecture for C/Fq is true, while in Section 2.2 we discuss when we can confirm that the Mertens conjecture for C/Fq is false. The results of these two sections combine to prove the following theorem.

Theorem 1.5. Let C be a nonsingular projective curve over Fq of genus g ≥ 1. Then the Mertens conjecture for C/Fq is false if the associated zeta function

ZC/Fq (u) has zeroes of multiple order. If ZC/Fq (u) has only simple zeroes, then the Mertens conjecture for C/Fq is true provided

X 1 γ ≤ 1, (1.5) Z 0 (γ−1) γ − 1 γ C/Fq where the sum is over the inverse zeroes γ of ZC/Fq (u). Furthermore, if C satisfies the Linear Independence hypothesis, then the converse is also true: the Mertens conjecture for C/Fq is true only when (1.5) holds. We remark that this does not entirely answer Question 1.2; it is possible that

C/Fq is such that (1.5) does not hold but that the Mertens conjecture for C/Fq is true; in order for this to happen, ZC/Fq (u) must only has simple zeroes but C must fail to satisfy the Linear Independence hypothesis. In Section 3.2, we give an example of a family of curves of genus one for which this occurs. Question 1.3 is the function field analogue of the problem of determining the logarithmic density of the set where the Mertens conjecture holds, as we discussed in Section 1.1. Section 2.3 deals with this question, where we are instead able to determine the natural density of the set of positive integers X for which (1.4) holds, under the proviso that the underlying curve satisfies the Linear Independence hypothesis; unlike the classical case, where this is conjectured to be true, this hypothesis can be violated for certain curves.

Theorem 1.6. Let C be a nonsingular projective curve over Fq of genus g ≥ 1, and suppose that C satisfies the Linear Independence hypothesis. The natural density

1  X/2 d PC/ q;µ = lim # 1 ≤ X ≤ Y : MC/ q (X) ≤ q F Y →∞ Y F 6 The Mertens Conjecture in Function Fields

exists and satisfies d PC/Fq;µ > 0, with d PC/Fq;µ = 1 if and only if

X 1 γ ≤ 1. Z 0 (γ−1) γ − 1 γ C/Fq
We in fact describe this density d PC/Fq;µ in terms of the Lebesgue measure of the pullback of a certain function of C/Fq, which allows us to determine this density exactly should we know the zeta function of the function field. In Chapter 3 we analyse Questions 1.2 and 1.3 in the low genus case g = 1, so that C/Fq is the function field of an elliptic curve over a finite field. We give an explicit classification of all elliptic curves satisfying the Mertens conjecture in terms of the order q of the finite field q and of the trace a of the Frobenius endomorphism acting on the elliptic curve C, in the form of the following theorem.

Theorem 1.7. Let C be an elliptic curve over a finite field Fq of characteristic p. Then the Mertens conjecture is true for C/Fq if and only if the order of the finite field q and the trace a of the Frobenius endomorphism acting on C over Fq satisfy precisely one of the following conditions:

(1) q = pm with a = 2, where either m is arbitrary and p 6= 2, or m = 1 and p = 2, √ (2) q = pm with a = q, where m is even and p 6≡ 1 (mod 3),

(3) q = pm with a = 0, where either m is even and p 6≡ 1 (mod 4), or n is odd.

In all these cases, we have that

MC/Fq (X) lim sup X/2 = 1. X→∞ q Furthermore, the natural density

1  X/2 d PC/ q;µ = lim # 1 ≤ X ≤ Y : MC/ q (X) ≤ q F Y →∞ Y F exists, and this density is equal to 1 if and only if q and a satisfy one of conditions (1)—(3).

Finally, for Question 1.4, we study in Chapter 4 the average proportion of curves in a certain family satisfying the Mertens conjecture as the finite field Fq grows larger. This allows us to use Deligne’s equidistribution theorem, a powerful result that links the average properties of curves to the Haar measure on certain 1.2 Mertens Conjectures in Function Fields 7

groups of random matrices. We choose to average over a certain family of curves, namely a family of hyperelliptic curves H2g+1,qn , for which most curves satisfy the Linear Independence hypothesis, where the notion of most curves satisfying a certain property is defined in Definition 4.3. Together with our resolution of Question 1.2, this allows us to relate the average proportion of hyperelliptic curves satisfying the Mertens conjecture to the Haar measure of the pullback of the region where a certain function of random matrices is at most 1. For low values of g, we may then calculate this Haar measure explicitly. Remarkably, we find that most curves in this family do not satisfy the Mertens conjecture.

Theorem 1.8. Fix 1 ≤ g ≤ 2, and suppose that the characteristic of Fq is odd. Then as n tends to infinity, most hyperelliptic curves C ∈ H2g+1,qn do not satisfy the Mertens conjecture for C/Fqn . The proof involves checking that a certain function in g variables on [0, π]g is bounded below by 1: for large g, this becomes very difficult. Nevertheless, it seems likely that this inequality holds for all g ≥ 1, as we indicate in Section 4.2, thereby leading us to formulate the following conjecture.

Conjecture 1.9. Fix g ≥ 1, and suppose that the characteristic of Fq is odd. Then as n tends to infinity, most hyperelliptic curves C ∈ H2g+1,qn do not satisfy the Mertens conjecture for C/Fqn . We end Chapter 4 by discussing two variations of Question 1.4 and showing how minor modifications of the proof of Theorem 1.8 lead to proofs of these new questions. These results build upon the work of Cha [5], who studies the closely related problem of determining the average size of

MC/Fq (X) lim sup X/2 X→∞ q over H2g+1,qn . Cha is able to show that a truncated form of this average converges to a certain integral over a particular space of random matrices, and by analysing this integral, Cha is led to conjecture the limiting behaviour of this average as the genus g tends to infinity. The purpose of this result is to formulate a function field analogue of a conjecture of Gonek (unpublished), which is studied by Ng in [21], stating that

M(x) M(x) 0 < lim sup √ 5/4 = − lim inf √ 5/4 < ∞. (1.6) x→∞ x (log log log x) x→∞ x (log log log x) 8 The Mertens Conjecture in Function Fields

While Cha’s results deviate in a different direction to the main results in this thesis, much of the groundwork is identical. We reproduce the proofs of many of these necessary results throughout this thesis, with attribution to Cha and identification of the location of the original proof in Cha’s paper [5]. Chapter 2

Local Mertens Conjectures

2.1 An Explicit Expression for MC/Fq (X)

In order to study the summatory function of the M¨obiusfunction of a function

field C/Fq, we must first introduce the associated zeta function ζC/Fq (s). Given a curve C over Fq of genus g, the zeta function ζC/Fq (s) is defined initially for <(s) > 1 by the absolutely convergent Dirichlet series

X 1 ζ (s) = , C/Fq N Ds D≥0 where the sum is over all effective divisors D of C, and N D = qdeg(D) is the −s absolute norm of D. Note that q , and hence ζC/Fq (s), is periodic with period

2πi/ log q. We also observe that much like the Riemann zeta function, ζC/Fq (s) has an Euler product for <(s) > 1,

Y 1 ζ (s) = , C/Fq 1 − N P −s P with the product over all prime divisors P of C. This in turn implies that ζC/Fq (s) is nonvanishing in the open half-plane <(s) > 1. More than this is true, however;

ζC/Fq (s) extends meromorphically to the entire complex plane.

Theorem 2.1 ([24, Theorem 5.9]). Given a nonsingular projective curve C over

Fq of genus g, there exists a polynomial PC/Fq (u) with integer coefficients of degree 2g such that for <(s) > 1,

P (q−s) ζ (s) = C/Fq . C/Fq (1 − q−s) (1 − q1−s)

9 10 Local Mertens Conjectures

This yields a meromorphic extension of ζC/Fq (s) to the whole complex plane, with simple poles at s = 2πik/ log q and s = 1+2πik/ log q for all k ∈ Z. Furthermore,

ζC/Fq (s) satisfies the functional equation

(g−1)s (g−1)(1−s) q ζC/Fq (s) = q ζC/Fq (1 − s).

2g The constant term of the polynomial PC/Fq (u) is 1, and the coefficient of u is g q . Finally, the value PC/Fq (1) is hC/Fq , the class number of C/Fq.

The polynomial PC/Fq (u) factorises over C as

2g Y PC/Fq (u) = (1 − γju) j=1

for some complex numbers γj, which we call the inverse zeroes of ζC/Fq (s). By the nonvanishing of ζC/Fq (s) outside of 0 ≤ <(s) ≤ 1 and the functional equation for ζC/Fq (s), we must have that 1 ≤ |γj| ≤ q. Moreover, the structure of the meromorphic continuation of ζC/Fq (s) to the entire complex plane shows that

ζC/Fq (s) = ζC/Fq (s). By this, we may conclude that the inverse zeroes γj must occur in reciprocal pairs; that is, we can order the inverse zeroes γj so that −1 γj+g = qγ for all 1 ≤ j ≤ g. Much more about the inverse zeroes is known; it j √ has been proven that they all have absolute value q.

Theorem 2.2 (Riemann Hypothesis for Function Fields [24, Theorem 5.10]). √ Each inverse zero γj of ζC/Fq (s) has absolute value q. Equivalently, all of the zeroes of ζC/Fq (s) lie along the line <(s) = 1/2. √ iθ(γj ) Consequently, we may write the inverse zeroes in the form γj = qe with √ −iθ(γj ) 0 ≤ θ(γj) ≤ π and γj+g = γj = qe for 1 ≤ j ≤ g. Note in particular that √ the orders of the inverse zeroes γ = ± q of ζC/Fq (s) must be even. −s Observe that ζC/Fq (s) is in fact a function of q . This allows us to define the −s zeta function ZC/Fq (u) via the identification u = q , so that

P (u) Z (u) = ζ (s) = C/Fq , (2.1) C/Fq C/Fq (1 − u)(1 − qu)

and hence ZC/Fq (u) satisfies the functional equation

 1  Z (u) = qg−1u2(g−1)Z . (2.2) C/Fq C/Fq qu 2.1 An Explicit Expression for MC/Fq (X) 11

Returning to the summatory function of the M¨obiusfunction of a function field, we now approach the problem of obtaining an explicit description of this function by studying the Dirichlet series

X µC/ q (D) F . N Ds D≥0

t As µC/Fq (D) is multiplicative and satisfies µC/Fq (P ) = −1 and µC/Fq (P ) = 0, t ≥ 2, for a prime divisor P of C, this Dirichlet series has the Euler product expansion

X µC/ q (D) Y F = 1 − N P −s
N Ds D≥0 P for <(s) > 1, which upon comparing Euler products leads us to the identity

X µC/ q (D) 1 F = , (2.3) N Ds ζ (s) D≥0 C/Fq which is valid for all <(s) > 1. On the other hand, note that for <(s) > 1, we may rearrange this Dirichlet series instead to be of the form

∞ X µC/ q (D) X µC/ q (D) X 1 X F = F = µ (D), (2.4) N Ds qdeg(D)s qNs C/Fq D≥0 D≥0 N=0 deg(D)=N and so if we can determine an expression for the coefficients of the Dirichlet se- ries for 1/ζC/Fq (s) using the known factorisation (2.1) of ζC/Fq (s), then upon comparing coefficients, we will be able to construct an accurate formula for PX−1 P MC/Fq (X) = N=0 deg(D)=N µC/Fq (D). This is particularly simple to do when g = 0, so that C is the projective line 1 P , and hence the function field C/Fq is simply Fq(t). In this case, we have that ∞ X N X 1 u µC/Fq (D) = = (1 − u)(1 − qu), ZC/ (u) N=0 deg(D)=N Fq and so by equating coefficients, we obtain the following result.

Proposition 2.3. Let g = 0. Then  1 if X = 1,  MC/Fq (X) = −q if X = 2,  0 if X ≥ 3.

In particular, the Mertens conjecture for C/Fq holds. 12 Local Mertens Conjectures

For g ≥ 1, our method for determining an expression for these coefficients is via Cauchy’s residue theorem. We will deal only with the case where all of the zeroes of ZC/Fq (u) are simple, though it is nevertheless possible to determine an explicit expression for MC/Fq (X) when ZC/Fq (u) has zeroes of multiple order [5, Proposition 2.2].

Proposition 2.4 (Cha [5, Proposition 2.2, Corollary 2.3]). Let g ≥ 1, and sup- pose that the zeroes of ZC/Fq (u) are all simple. Then as X tends to infinity,   MC/ q (X) X 1 γ 1 F = − eiXθ(γ) + O . (2.5) qX/2 Z 0 (γ−1) γ − 1 q,g qX/2 γ C/Fq

In particular, the quantity

MC/Fq (X) B(C/Fq) = lim sup X/2 X→∞ q satisfies

X 1 γ B(C/ q) ≤ . (2.6) F Z 0 (γ−1) γ − 1 γ C/Fq It is useful to compare the explicit expression (2.5) to that for the classical case, (1.3). One can immediately see the similarities, with the chief difference being the replacement of x in the classical setting by qX for the function field case.

Proof. This is proved by Cha in [5, Proposition 2.2]; we include the details of the T proof for later comparison. Let CT = {z ∈ C : |z| = q } for T > 0, and consider the contour integral 1 I 1 1 N+1 du. 2πi CT u ZC/Fq (u)

We can write 1/ZC/Fq (u) in two ways; via (2.3) and (2.1), and via (2.4), yielding the identities

2g 1 Y 1 = (1 − u)(1 − qu) , (2.7) Z (u) 1 − γ u C/Fq j=1 j ∞ 1 X N X = u µC/Fq (D), (2.8) ZC/ (u) Fq N=0 deg(D)=N

−1 −1 where the first identity is valid for all u ∈ C \{γ1 , . . . , γ2g }, and the second −1 identity is valid for all |u| < q . So the singularities of the integrand inside CT 2.1 An Explicit Expression for MC/Fq (X) 13

−1 −1 occur at u = 0 and at u = γ for each zero γ of ZC/Fq (u). At the singularity u = 0, we have by (2.8) that

1 1 X Res N+1 = µC/Fq (D). u=0 u ZC/ (u) Fq deg(D)=N

−1 As ZC/Fq (u) has a simple zero at each γ , we obtain from (2.7) that

−1 1 1 1 u − γ 1 N+1 Res = lim = 0 γ . u=γ−1 N+1 u→γ−1 N+1 −1 u ZC/Fq (u) u ZC/Fq (u) ZC/Fq (γ )

So by Cauchy’s residue theorem, I 1 1 1 X 1 N+1 X N+1 du = 0 −1 γ + µC/Fq (D). (2.9) 2πi u ZC/ (u) ZC/ (γ ) CT Fq γ Fq deg(D)=N

Summing over all 0 ≤ N ≤ X − 1 and evaluating the resulting geometric series, we find that

X 1 γ M (X) = − γX + R (q, g, T ), (2.10) C/Fq Z 0 (γ−1) γ − 1 X γ C/Fq where the error term RX (q, g, T ) is

X−1 X 1 γ X 1 I 1 1 R (q, g, T ) = + du. (2.11) X Z 0 (γ−1) γ − 1 2πi uN+1 Z (u) γ C/Fq N=0 CT C/Fq

T √ Now (2.7) and the fact that |u| = q and |γj| = q imply that

I I 1 1 1 1 1 1 N+1 du ≤ N+1 |du| 2πi CT u ZC/Fq (u) 2π CT u ZC/Fq (u) qT + 1
q1+T + 1
≤ q−NT . (q1/2+T − 1)2g

As the right-hand side of (2.9) is independent of T , we may take the limit as T tends to infinity in order to find that the contour integral above is zero if N ≥ 3 − 2g and at most q1−g in absolute value if N = 2(1 − g). As g ≥ 1 and

N ≥ 0, this implies that for all X ≥ 1, the second term in RX (q, g, T ) vanishes if g ≥ 2, and is a constant of absolute value at most 1 if g = 1. Thus RX (q, g, T ) is constant, and hence bounded as X tends to infinity. Upon dividing through √ (2.10) by qX/2 and using the fact that γ = qeiθ(γ), we obtain the result. 14 Local Mertens Conjectures

As there are precisely 2g zeroes of ZC/Fq (u), the sum in (2.6) is finite, and X/2 hence MC/Fq (X)/q is bounded. Furthermore, (2.6) proves part of Theorem 1.5 in showing that if ZC/Fq (u) has simple zeroes, then the inequality (1.5) holding implies that the Mertens conjecture for C/Fq is true.

We next show that the bound (2.6) is sharp if the zeroes of ZC/Fq (u) are particularly well-behaved.

Definition 2.5. We say that C satisfies the Linear Independence hypothesis, which we abbreviate to LI, if the collection

π, θ(γ1), . . . , θ(γg) is linearly independent over the rational numbers.

Notably, if C satisfies LI, then all of the zeroes of ZC/Fq (u) are simple. Theorem 2.6 (Cha [5, Theorem 2.5]). Suppose that C satisfies LI. Then

X 1 γ B(C/ q) = . F Z 0 (γ−1) γ − 1 γ C/Fq

Consequently, if C satisfies LI, then the Mertens conjecture for C/Fq is true if and only if the inequality (1.5) holds.

The proof follows from a direct application of the Kronecker–Weyl theorem, which we prove in Appendix A in the following form.

Lemma 2.7 (Kronecker–Weyl Theorem). Let t1, . . . , tg be real numbers, and let H be the topological closure in the g-torus

g g T = {(z1, . . . , zg) ∈ C : |zj| = 1 for all 1 ≤ j ≤ g}. of the subgroup

 2πiXt1 2πiXtg
g He = e , . . . , e ∈ T : X ∈ Z .

g Then H is a closed subgroup of T . In particular, when the collection 1, t1, . . . , tg is linearly independent over the rational numbers, H is precisely Tg. Furthermore, g for arbitrary t1, . . . , tg and for any continuous function h : T → C, we have that

1 Y Z X 2πiXt1 2πiXtg
lim h e , . . . , e = h(z) dµH (z), Y →∞ Y X=1 H where µH is the normalised Haar measure on H. 2.1 An Explicit Expression for MC/Fq (X) 15

Proof of Theorem 2.6. As γj+g = γj for each 1 ≤ j ≤ g and as ZC/Fq (u) =

ZC/Fq (u), X 1 γ eiXθ(γ) Z 0 (γ−1) γ − 1 γ C/Fq g ! X 1 γj 1 γj = eiXθ(γj ) + e−iXθ(γj ) Z 0 γ−1
γ − 1 Z 0 (γ −1) γ − 1 j=1 C/Fq j j C/Fq j j g ! X 1 γj iXθ(γ ) 1 γj = e j + eiXθ(γj ) Z 0 γ−1
γ − 1 Z 0 γ−1
γ − 1 j=1 C/Fq j j C/Fq j j g ! X 1 γj = 2< eiXθ(γj ) . Z 0 γ−1
γ − 1 j=1 C/Fq j j

X/2 Thus we may write MC/Fq (X)/q as g !   MC/ q (X) X 1 γj 1 F = −2< eiXθ(γj ) + O . qX/2 Z 0 γ−1
γ − 1 q,g qX/2 j=1 C/Fq j j The assumption that C satisfies LI then allows us to apply the Kronecker–Weyl theorem with tj = θ(γj)/2π for 1 ≤ j ≤ g, which tells us that the set

 iXθ(γ1) iXθ(γg)
g e , . . . , e ∈ T : X ∈ N is dense (in fact, equidistributed) in Tg. This implies the existence of a subse- quence (Xm) of N such that

lim eiXmθ(γ1), . . . , eiXmθ(γg)
= e−iω(γ1), . . . , e−iω(γg)
, m→∞ where for 1 ≤ j ≤ g, ! 1 γj ω(γj) = arg − 0 −1
. ZC/Fq γj γj − 1 Together with (2.6), this implies that

g MC/ q (X) X 1 γj lim sup F = 2 . X/2 0 −1
X→∞ q Z γ γ − 1 j=1 C/Fq j j An analogous argument shows that

g MC/ q (X) X 1 γj lim inf F = −2 . X→∞ qX/2 Z 0 γ−1
γ − 1 j=1 C/Fq j j 16 Local Mertens Conjectures

2.2 MC/Fq (X) and Zeroes of Multiple Order

To complete the proof of Theorem 1.5, it remains to consider the case where

ZC/Fq (u) has zeroes of multiple order; we will show that in this situation, the Mertens conjecture can never hold. We first require the following trivial bound on MC/Fq (X).

Lemma 2.8. For any nonsingular curve C over Fq of genus g ≥ 1, we have that

X
MC/Fq (X) = Oq,g q . (2.12)

Proof. By taking absolute values, we trivially have that

X−1 X MC/Fq (X) ≤ bC/Fq (N), N=0 where bC/Fq (N) is the number of effective divisors of degree N. From [24, Lemma 5.8], there exists a constant c dependent on C/Fq such that

N bC/Fq (N) ∼ cq

for all N > 2g − 2, while bC/Fq (N) is finite for 0 ≤ N ≤ 2g − 2. Summing over all 0 ≤ N ≤ X − 1 yields the result.

Corollary 2.9. For |u| < q−1,

∞ 1 X = (1 − u) M (X)uX−1. (2.13) Z (u) C/Fq C/Fq X=1

Proof. Via partial summation, we have that for |u| < q−1 and for Y ≥ 1 that

Y −1 Y −1 X N X Y −1 X X X−1
u µC/Fq (D) = MC/Fq (Y )u − MC/Fq (X) u − u . N=0 deg(D)=N X=1

By taking the limit as Y tends to infinity and using (2.3) and (2.12), we obtain the result.

The key result that we make use of is the following.

Lemma 2.10 (Landau’s Theorem). Let A(X) be real-valued sequence, and sup- pose that there exists a positive integer X0 such that A(X) is of constant sign 2.2 MC/Fq (X) and Zeroes of Multiple Order 17

for all X ≥ X0. Furthermore, suppose that the supremum vc of the set of points v ∈ [0, ∞) for which the sum

∞ X A(X)vX−1

X=X0 converges satisfies vc ≤ 1. Then the function

∞ X F (u) = A(X)uX−1 X=1 is holomorphic in the disc |u| < vc with a singularity at the point vc.

Proof. By making the change of variables v = e−σ, we have that

∞ σ Z ∞ X X−1 σe −Xσ A(X)v = − −σ A (bXc) e dX 1 − e X0 X=X0 σeσ Z ∞ A (blog xc) dx = − −σ σ , 1 − e eX0 x x where bXc denotes the integer part of X, and the second equality follows from X −s the substitution x = e . Similarly, letting u = e for <(s) > vc, we have that

∞ X ses Z ∞ A (blog xc) dx A(X)uX−1 = − . 1 − e−s xs x X=1 e The result now follows directly from [19, Lemma 15.1].

Finally, we also require the following combinatorial identity.

Lemma 2.11. Let |u| < 1, and let r be a positive integer. Then

∞ r−1 X 1 X Xr−1uX = A(r − 1, k)uk+1, (2.14) (1 − u)r X=1 k=0 where the coefficients

k X r A(r − 1, k) = (−1)j(k + 1 − j)r−1 j j=0 are Eulerian numbers, which satisfy the identity

r−1 X A(r − 1, k) = r!. k=0 18 Local Mertens Conjectures

√ We first deal with the case where q is an inverse zero of ZC/Fq (u). √ Proposition 2.12. Let g ≥ 1, and suppose that γ = q is an inverse zero of

ZC/Fq (u). Then γ has order r ≥ 2, and

MC/Fq (X) lim sup r−1 X/2 > 0. X→∞ X q

In particular, the Mertens conjecture for the function field C/Fq is false. √ Proof. If u = q is an inverse zero of ZC/Fq (u), then this zero must be of order r ≥ 2 due to the functional equation for ZC/Fq (u). In this case,

r u − q−1/2
r! lim = (r) , u→q−1/2 Z (u) −1/2 C/Fq ZC/Fq (q ) and this is nonzero and real as ZC/Fq (v) is real for all real v, so all derivatives of any order of ZC/Fq (v) at real values v must be real. r (r) −1/2
Now if (−1) /ZC/Fq q is negative, then we suppose that there exists r−1 X/2 some c ≥ 0 and a positive integer X0 such that MC/Fq (X) > −cX q for all X ≥ X0; we will show that for this to be the case, we must have that c ≥ c0 for a certain c0 > 0, and hence that

MC/Fq (X) lim inf ≤ −c0 < 0. X→∞ Xr−1qX/2

r−1 X/2 Indeed, if MC/Fq (X) > −cX q for all X ≥ X0, then by (2.13) and (2.14),

∞ X r−1 X/2
X−1 MC/Fq (X) + cX q u X=1 r−1 1 c X (k+1)/2 k+1 = + √
r A(r − 1, k)q u . (2.15) (1 − u)ZC/ (u) u 1 − qu Fq k=0

The right-hand side of (2.15) is holomorphic for |u| < q−1/2 and has a singularity at u = q−1/2, so Landau’s theorem implies that the sum on the left-hand side of (2.15) converges for all |u| < q−1/2 and defines a holomorphic function F (u) on √ r this open half-plane. We then multiply both sides of (2.15) by 1 − qu
and consider the limit as u tends to q−1/2 from the left through real values; from the right-hand side of (2.15), we find that this limit exists and is equal to

(−1)rqr/2r! √ + c qr!. −1/2 (r) −1/2 (1 − q ) ZC/Fq (q ) 2.2 MC/Fq (X) and Zeroes of Multiple Order 19

Now if this were negative, then the left-hand side of (2.15) would tend to negative infinity as u approaches q−1/2 from the left. This, however, is impossible, as we can split up this sum into two parts: a sum from X = 1 to X0 − 1, and a sum from X = X0 to infinity, and the former sum is uniformly bounded as u tends to q−1/2, while the coefficients of the latter sum are nonnegative. Consequently, we r−1 X/2 conclude that the inequality MC/Fq (X) > −cX q for all X ≥ X0 can only hold provided (−1)r+1qr/2 c ≥ > 0. √
(r) −1/2 q − 1 ZC/Fq (q ) r (r) If (−1) /ζC/Fq (1/2) is positive, on the other hand, we instead suppose that r−1 X/2 the inequality MC/Fq (X) < cX q holds for all X ≥ X0, in which case an analogous argument applied to the equation

∞ X r−1 X/2
X−1 MC/Fq (X) − cX q u X=1 r−1 1 c X (k+1)/2 k+1 = − √
r A(r − 1, k)q u (1 − u)ZC/ (u) u 1 − qu Fq k=0 shows that r r/2 MC/Fq (X) (−1) q lim sup ≥ √ > 0. Xr−1qX/2
(r) −1/2 X→∞ q − 1 ZC/Fq (q ) −1/2
The case when ZC/Fq q 6= 0 but ZC/Fq (u) nevertheless has a zero of multiple order follows by a similar but slightly more complicated argument.

Proposition 2.13. Let g ≥ 1, and suppose that ZC/ (u) has an inverse zero √ Fq √ γ = qeiθ(γ) of order r ≥ 2, but that the order of the inverse zero at q is of order strictly less than r. Then

MC/Fq (X) lim sup r−1 X/2 > 0. X→∞ X q

In particular, the Mertens conjecture for the function field C/Fq is false.

Proof. Suppose there exists some c ≥ 0 and a positive integer X0 such that r−1 X/2 MC/Fq (X) > −cX q for all X ≥ X0. Once again, Landau’s theorem shows that the equation

∞ X r−1 X/2
X−1 MC/Fq (X) + cX q u X=1 r−1 1 c X (k+1)/2 k+1 = + √
r A(r − 1, k)q u . (1 − u)ZC/ (u) u 1 − qu Fq k=0 20 Local Mertens Conjectures

is valid for |u| < q−1/2 and defines a holomorphic function F (u) in this disc. Then for |u| < q−1/2,

∞ X X/2 r−1
X−1 MC/Fq (X) + cq X (1 + cos (φ(γ) − (X − 1)θ(γ))) u X=1 eiφ(γ) 1 − ue−iθ(γ) e−iφ(γ) 1 − ueiθ(γ) = F (u) + F ue−iθ(γ)
+ F ueiθ(γ)
, (2.16) 2 1 − u 2 1 − u where we let ! (−1)rγr φ(γ) = π − arg . (r) −1 ZC/Fq (γ ) √ r Upon multiplying both sides of (2.16) by 1 − qu
, we find via the right-hand side of (2.16) that as u tends to q−1/2 from the left through real values, this quantity converges to

√ qr/2r! c qr! − , (1 − q−1/2) Z (r) (γ−1) C/Fq which must be nonnegative: otherwise, the left-hand side of (2.16) would tend to negative infinity as u approaches q−1/2 from the left, a contradiction given the −1/2 uniform boundedness of the sum from X = 0 to X0 as u tends to q and the fact that the sum from X = X0 to infinity is nonnegative. Thus qr/2 c ≥ √ > 0, q − 1
Z (r) (γ−1) C/Fq and so M (X) qr/2 lim inf C/Fq ≤ − < 0. r−1 X/2 √ X→∞ X q q − 1
Z (r) (γ−1) C/Fq An analogous argument shows that we also have that

M (X) qr/2 lim sup C/Fq ≥ > 0. r−1 X/2 √ X→∞ X q q − 1
Z (r) (γ−1) C/Fq

X/2 2.3 The Limiting Distribution of MC/Fq (X)/q

Let m denote the Lebesgue measure on [0, 1]g. For a Borel set B ⊂ R and a g Borel-measurable function f : [0, 1] → R, we write m(f(θ1, . . . , θg) ∈ B) for

g m ({(θ1, . . . , θg) ∈ [0, 1] : f(θ1, . . . , θg) ∈ B}) . X/2 2.3 The Limiting Distribution of MC/Fq (X)/q 21

Our main result for this section is the following expression for the natural density X/2 of the set of positive integers X for which MC/Fq (X) ≤ q , the proof of which is similar to that of the key result of Cha in [4] on Chebyshev’s bias in function fields, which in turn is based on the seminal work of Rubinstein and Sarnak in [25].

Proposition 2.14. Let C be a nonsingular projective curve over Fq of genus g ≥ 1, and suppose that C satisfies LI. The natural density

1  X/2 d PC/ q;µ = lim # 1 ≤ X ≤ Y : MC/ q (X) ≤ q F Y →∞ Y F exists and is equal to

g ! X 1 γj d P
= m −1 ≤ 2 cos(2πθ ) ≤ 1 . (2.17) C/Fq;µ Z 0 γ−1
γ − 1 j j=1 C/Fq j j

From this, the proof of Theorem 1.6 follows quite easily: it is clear that this density is strictly positive, as there exists an open neighbourhood of the point (1/4,..., 1/4) ∈ [0, 1]g such that

g X 1 γj −1 ≤ 2 cos(2πθ ) ≤ 1 Z 0 γ−1
γ − 1 j j=1 C/Fq j j
inside this neighbourhood, while it is immediate that d PC/Fq;µ = 1 when

X 1 γ ≤ 1. Z 0 (γ−1) γ − 1 γ C/Fq
If the inequality above does not hold, however, then d PC/Fq;µ < 1, for then there exists an open neighbourhood of (0,..., 0) ∈ [0, 1]g such that

g X 1 γj 2 cos(2πθ ) > 1 Z 0 γ−1
γ − 1 j j=1 C/Fq j j inside this neighbourhood. In fact, we prove something slightly more general than Proposition 2.14: we X/2 show that MC/Fq (X)/q has a limiting distribution as X tends to infinity, the construction of which is based off the Kronecker–Weyl theorem. For any nonsin- gular projective curve C over Fq of genus g ≥ 1, (2.5) allows us to write M (X) C/Fq = E (X) + ε (X), qX/2 C/Fq;µ C/Fq;µ 22 Local Mertens Conjectures

where X 1 γ E (X) = − eiXθ(γ), C/Fq;µ Z 0 (γ−1) γ − 1 γ C/Fq  1  ε (X) = O , C/Fq;µ q,g qX/2

−1 provided that all of the zeroes γ of ZC/Fq (u) are simple. We begin by first constructing the limiting distribution of EC/Fq;µ(X).

Lemma 2.15. Let C be a nonsingular projective curve over Fq of genus g ≥ 1, −1 and suppose that all of the zeroes γ of ZC/Fq (u) are simple. There exists a probability measure νC/Fq;µ on R that satisfies Y Z 1 X
lim f EC/ ;µ(X) = f(x) dνC/ ;µ(x) Y →∞ Y Fq Fq X=1 R for all continuous functions f on R.

Proof. By the Kronecker–Weyl theorem with tj = θ(γj)/2π for 1 ≤ j ≤ g, there exists a subtorus H ⊂ Tg satisfying 1 Y Z X iXθ(γ1) iXθ(γg)
lim h e , . . . , e = h(z) dµH (z) Y →∞ Y X=1 H g for every continuous function h on T , where µH is the normalised Haar measure on H. We now define the probability measure νC/Fq;µ on R by

νC/Fq;µ(B) = µH (Be) for each Borel set B ⊂ R, where ( g ! ) X 1 γj Be = (z1, . . . , zg) ∈ H : −2< zj ∈ B . Z 0 γ−1
γ − 1 j=1 C/Fq j j The function g ! X 1 γj −2< z Z 0 γ−1
γ − 1 j j=1 C/Fq j j is continuous on H, so Be is a Borel set in H, and νC/Fq;µ is a probability measure as µH is the normalised Haar measure on H. So for a bounded continuous function g f on R, we define the function h(z1, . . . , zg) on the g-torus T by g !! X 1 γj h(z , . . . , z ) = f −2< z , 1 g Z 0 γ−1
γ − 1 j j=1 C/Fq j j X/2 2.3 The Limiting Distribution of MC/Fq (X)/q 23

so that h is continuous on Tg with

iXθ(γ1) iXθ(γg)
f EC/Fq;µ(X) = h e , . . . , e .

Hence by the Kronecker–Weyl theorem, Z Z f(x) dνC/Fq;µ(x) = h(z1, . . . , zg) dµH (z1, . . . , zg) R H Y 1 X = lim h eiXθ(γ1), . . . , eiXθ(γg)
Y →∞ Y X=1 Y 1 X
= lim f EC/ ;µ(X) . Y →∞ Y Fq X=1

X/2 Next, we show using this construction that MC/Fq (X)/q has a limiting distribution on R. The key tool is the following result that allows us to show that a sequence of measures is weakly convergent.

∞ Lemma 2.16 (Portmanteau Theorem [2, Theorem 2.1]). Let {νY }Y =1, ν be prob- ability measures on a metric space X . Then the following are equivalent:

(1) The sequence of measures νY converges weakly to ν; that is, Z Z lim f(x) dνY (x) = f(x) dν(x). Y →∞ X X for every bounded continuous function f : X → R.

(2) For every Borel set B ⊂ X whose boundary has ν-measure zero,

lim νY (B) = ν(B). Y →∞

(3) For every bounded Lipschitz continuous function f : X → R, Z Z lim f(x) dνY (x) = f(x) dν(x). Y →∞ X X

We also require the following lemma to show the existence of a weak limit of measures. This relies on the notion of tightness of a sequence of measures: we say that a family of probability measures {νY } on a metric space X is tight if for every ε > 0 there exists a compact set K ⊂ X such that νY (K) > 1 − ε for all Y .

∞ Lemma 2.17 (Prohorov’s Theorem [2, Theorem 5.1]). Let {νY }Y =1 be probability ∞ measures on a metric space X . If {νY }Y =1 is tight, then every subsequence of ∞ {νY }Y =1 has a weakly convergent subsubsequence. 24 Local Mertens Conjectures

Proposition 2.18. Let C be a nonsingular projective curve over Fq of genus −1 g ≥ 1, and suppose that all of the zeroes γ of ZC/Fq (u) are simple. The function X/2 MC/Fq (X)/q has a limiting distribution νC/Fq;µ on R. That is, there exists a probability measure νC/Fq;µ on R such that

Y   Z 1 X MC/Fq (X) lim f = f(x) dνC/ ;µ(x) Y →∞ Y qX/2 Fq X=1 R for all bounded continuous functions f on R.

Proof. For each positive integer Y , let νY,µ be the probability measure on R given by 1  M (X)  ν (B) = # 1 ≤ X ≤ Y : C/Fq ∈ B Y,µ Y qX/2 for any Borel set B ⊂ R, so that for any continuous function f on R, Z Y   1 X MC/ q (X) f(x) dν (x) = f F . Y,µ Y qX/2 R X=1

X/2 As MC/Fq (X)/q is bounded, the probability measures {νY,µ} are tight, so by Prohorov’s Theorem, for every subsequence {Yk} there exists a subsubsequence

{Yk } and a probability measure νC/ ;µ such that νY ;µ converges weakly to ` e Fq k` ν . We will show that ν = ν for every such subsequence, which eC/Fq;µ eC/Fq;µ C/Fq;µ will imply that the probability measures {νY,µ} converge weakly to νC/Fq;µ, as required.

So if νY ;µ converges weakly to νC/ ;µ, then by the Portmanteau theorem, k` e Fq

Y k`   Z Z 1 X MC/Fq (X) lim f = lim f(x) dνY ;µ(x) = f(x) dνC/ ;µ(x) X/2 k` e Fq `→∞ Yk q `→∞ ` X=1 R R for every bounded Lipschitz continuous function f : R → R, that is, for every function f for which there exists a constant cf ≥ 0 such that

|f(x) − f(y)| sup = cf < ∞. x,y∈R |x − y| x6=y

The Lipschitz condition implies that

Y Y Y k`   k` k` 1 X MC/Fq (X) 1 X
cf X f X/2 ≥ f EC/Fq;µ(X) − |εC/Fq;µ(X)|. Yk q Yk Yk ` X=1 ` X=1 ` X=1 X/2 2.3 The Limiting Distribution of MC/Fq (X)/q 25

As ` tends to infinity, the left-hand side converges to R f(x) dν (x) by as- eC/Fq;µ R R sumption. On the right-hand side, the first term converges to f(x) dνC/ ;µ(x) R Fq −X/2 by Lemma 2.15, while the second term tends to zero as εC/Fq;µ(X) = O(q ). Thus Z Z f(x) dν (x) ≥ f(x) dν (x). eC/Fq;µ C/Fq;µ R R Furthermore, the Lipschitz condition also implies that

Y Y Y k`   k` k` 1 X MC/Fq (X) 1 X
cf X f X/2 ≤ f EC/Fq;µ(X) + |εC/Fq;µ(X)|, Yk q Yk Yk ` X=1 ` X=1 ` X=1 and hence Z Z f(x) dν (x) ≤ f(x) dν (x). eC/Fq;µ C/Fq;µ R R Combining both inequalities shows that νY ;µ converges weakly to νC/ ;µ. By k` Fq the uniqueness of weak limits of measures, we conclude that ν = ν . eC/Fq;µ C/Fq;µ Proof of Proposition 2.14. The Portmanteau Theorem together with Proposition 2.18 implies that   1 MC/Fq (X) lim # 1 ≤ X ≤ Y : ∈ B = νC/ ;µ(B) Y →∞ Y qX/2 Fq for every Borel set B ⊂ R whose boundary has νC/Fq;µ-measure zero. From
this, we can show that d PC/Fq;µ exists and is equal to νC/Fq;µ([−1, 1]) provided

νC/Fq;µ({−1, 1}) = 0. To prove this last point, we observe that the assumption that C satisfies LI implies that the topological closure of

 iθ(γ1)X iθ(γg)X
g He = e , . . . , e ∈ T : X ∈ Z in Tg is H = Tg. So the normalised Haar measure on H is the Lebesgue measure on the g-torus, and consequently for a Borel set B ⊂ R,

g ! ! X 1 γj ν (B) = m −2< e2πiθj ∈ B C/Fq;µ Z 0 γ−1
γ − 1 j=1 C/Fq j j g ! X 1 γj = m 2 cos(2πθ ) ∈ B Z 0 γ−1
γ − 1 j j=1 C/Fq j j by the translation invariance of the Lebesgue measure. Note that

g X 1 γj 2 cos(2πθ ) Z 0 γ−1
γ − 1 j j=1 C/Fq j j 26 Local Mertens Conjectures

is real analytic on [0, 1]g and not uniformly constant. As the zero set of a non- uniformly zero real analytic function has Lebesgue measure zero, we determine that g ! X 1 γj m 2 cos(2πθ ) = c = 0 Z 0 γ−1
γ − 1 j j=1 C/Fq j j
for any c ∈ R. Thus νC/Fq;µ is atomless, and hence d PC/Fq;µ is equal to

νC/Fq;µ([−1, 1]). Chapter 3

Examples in Low Genus

3.1 Elliptic Curves over Finite Fields

In this chapter, we study local Mertens conjectures in the simplest nontrivial case, namely g = 1, where we suppose that C/Fq is the function field of an elliptic curve over a finite field. That is, we suppose that C is a nonsingular projective algebraic curve of genus one over Fq with a given point defined over Fq. Then ZC/Fq (u) is of the form (1 − γu)(1 − γu) 1 − au + qu2 Z (u) = = C/Fq (1 − u)(1 − qu) (1 − u)(1 − qu) √ for some γ = qeiθ(γ) with 0 ≤ θ(γ) ≤ π, so that the integer a satisfies √ a = 2<(γ) = 2 q cos θ(γ).

Equivalently, γ can be defined in terms of the integer a via

 a  θ(γ) = arccos √ . 2 q

Geometrically, the integer a is the trace of the Frobenius endomorphism acting on the elliptic curve C over Fq. Notably, there are several restrictions on the possible values that a may take. The following lemma fully characterises the possible values of a.

Lemma 3.1 (Waterhouse [29, Theorem 4.1]). Let a be an integer. Then a is the trace of the Frobenius endomorphism acting on some elliptic curve C over a

finite field Fq of characteristic p if and only if one of the following conditions is satisfied:

27 28 Examples in Low Genus

√ (1) a 6≡ 0 (mod p) and |a| < 2 q; for such an integer a, the corresponding angle θ(γ) is such that θ(γ)/π is irrational,

√ (2) (i) q = pm with a = 2 q, where m is even, so that θ(γ) = 0,

√ (2) (ii) q = pm with a = −2 q, where m is even, so that θ(γ) = π,

√ (3) (i) q = pm with a = q, where m is even and p 6≡ 1 (mod 3), so that θ(γ) = π/3,

√ (3) (ii) q = pm with a = − q, where m is even and p 6≡ 1 (mod 3), so that θ(γ) = 2π/3, √ (4) (i) q = 2m with a = 2q, where m is odd, so that θ(γ) = π/4, √ (4) (ii) q = 2m with a = − 2q, where m is odd, so that θ(γ) = 3π/4, √ (4) (iii) q = 3m with a = 3q, where m is odd, so that θ(γ) = π/6, √ (4) (iv) q = 3m with a = − 3q, where m is odd, so that θ(γ) = 5π/6,

(5) q = pm with a = 0, where either m is even and p 6≡ 1 (mod 4), or m is odd, so that θ(γ) = π/2.

From the second part of this lemma, we may completely determine which elliptic curves satisfy LI.

Corollary 3.2. Let C be an elliptic curve over a finite field Fq of characteristic p, so that a and q satisfying one of conditions (1)—(5) of Lemma 3.1. Then C satisfies LI if and only if condition (1) is satisfied, ZC/Fq (u) has zeroes of multiple order if and only if condition (2) is satisfied, and C fails to satisfy LI but ZC/Fq (u) has only simple zeroes if and only if one of conditions (3)—(5) is satisfied.

X/2 Next, we determine an explicit expression for MC/Fq (X)/q using Propo- sition 2.4; remarkably, we may eliminate any error term for this expression.

We must consider two cases: when ZC/Fq (u) has only simple zeroes, and when

ZC/Fq (u) has a zero of order 2. For the first case, we have the following result.

Proposition 3.3. Let C be an elliptic curve over Fq, and suppose that ZC/Fq (u) 3.1 Elliptic Curves over Finite Fields 29

has only simple zeroes. Then M (X) rq + 1 − a C/Fq = 2 cos (ω + Xθ) , (3.1) qX/2 4q − a2 where a is the trace of the Frobenius endomorphism, and ω ∈ (−π/2, π/2), θ ∈ [0, π] are given by ! a − 2 ω = arctan , (3.2) 2p4q − a2  a  θ = arccos √ . (3.3) 2 q We remark that (3.1) is equivalent to M (X) a − 2 C/Fq = cos(Xθ) − sin(Xθ). (3.4) qX/2 p4q − a2

Proof. The fact that ZC/Fq (u) has only simple zeroes is equivalent to γ 6= γ. Now using the fact that γ(1 − γγ−1) γ γ − γ Z 0 γ−1
= − = − , C/Fq (1 − γ−1)(1 − qγ−1) γ − 1 γ − 1 we find from (2.10) that M (X) γ − 1 γ − 1 1 C/Fq = eiXθ(γ) + e−iXθ(γ) + R (q, 1,T ) qX/2 γ − γ γ − γ qX/2 X  γ − 1  1 = 2< eiXθ(γ) + R (q, 1,T ), γ − γ qX/2 X

√ iθ(γ) with RX (q, 1,T ) as in (2.11). As γ = qe , we see that √ √ √ γ − 1 q cos θ(γ) − 1 − i q sin θ(γ) pq + 1 − 2 q cos θ(γ) = √ = √ eiω(γ), γ − γ −2i q sin θ(γ) 2 q sin θ(γ) where √  q cos θ(γ) − 1 ω(γ) = arctan √ , q sin θ(γ) and consequently p √ MC/ (X) q + 1 − 2 q cos θ(γ) 1 Fq = √ cos (ω(γ) + Xθ(γ)) + R (q, 1,T ) qX/2 q sin θ(γ) qX/2 X √ q cos θ(γ) − 1 1 = cos(Xθ(γ)) − √ sin(Xθ(γ)) + R (q, 1,T ), q sin θ(γ) qX/2 X (3.5) 30 Examples in Low Genus

where the second equality follows from the cosine angle-sum formula. Now the proof of Proposition 2.4 shows that RX (q, g, T ) is constant for X ≥ 3 − 2g, and hence for all X ≥ 1 when g = 1. We can therefore determine the value of

RX (q, 1,T ) simply by taking X = 1 in (3.5), so that

MC/Fq (1) = 1 + R1(q, 1,T ).

On the other hand, X MC/Fq (1) = µC/Fq (D) = 1 deg(D)=0 as the only divisor of degree zero is the zero divisor, and so RX (q, 1,T ) = 0. We √ complete the proof by noting that a = 2 q cos θ(γ) with 0 ≤ θ(γ) ≤ π, so that √ p 2 q sin θ(γ) = 4q − a2.

The analogous result in the case where ZC/Fq (u) has a zero of multiple order is the following.

Proposition 3.4. Let C be an elliptic curve over a finite field Fq of characteristic m p, and suppose that ZC/ (u) has zeroes of multiple order, so that q = p with √ Fq a = ±2 q, where m is even. Then   MC/ (X) 1 Fq = −(±1)X 1 ∓ √ X + (±1)X . (3.6) qX/2 q √ √ Proof. If a = ±2 q, then γ = γ = ± q. From the proof of Proposition 2.4, we have that for N ≥ 0,

X 1 1 1 I 1 1 µC/Fq (D) = − Res N+1 + N+1 du, u=±q−1/2 u ZC/ (u) 2πi u ZC/ (u) deg(D)=N Fq CT Fq with the last term equal to zero for N ≥ 1. Now

2 1 1 d u ∓ q−1/2
(1 − u)(1 − qu) Res = lim −1/2 N+1 −1/2 N+1 √
2 u=±q u ZC/Fq (u) u→±q du u 1 ∓ qu √ = (±1)N+1 ( q ∓ 1)2 Nq(N−1)/2.

P This vanishes when N = 0, whereas deg(D)=0 µC/Fq (D) = 1, and so 1 I 1 1 du = 1. 2πi CT u ZC/Fq (u) 3.2 Proof of Theorem 1.7 31

Consequently,  X N+1 √ 2 (N−1)/2 1 if N = 0, µC/Fq (D) = −(±1) ( q ∓ 1) Nq + deg(D)=N 0 otherwise, which leads to the result upon summing over all 0 ≤ N ≤ X −1 and then dividing through by qX/2.

These two results can now be used to find the values

MC/Fq (X) B(C/Fq) = lim sup X/2 , X→∞ q
1  X/2 d PC/ q;µ = lim # 1 ≤ X ≤ Y : MC/ q (X) ≤ q F Y →∞ Y F for each elliptic curve C over a given finite field Fq. In the following section, we determine these two values for each possible combination of values for q and a as determined in Lemma 3.1, culminating in a proof of Theorem 1.7.

3.2 Proof of Theorem 1.7

We must determine B(C/Fq) and d PC/Fq;µ for the restricted values of q and a found in conditions (1)—(5) of Lemma 3.1.

√ (1) If q = pm with a 6≡ 0 (mod p) and |a| < 2 q, then θ/π is irrational, with θ as in (3.3). The Kronecker–Weyl theorem then shows that Xθ is equidistributed modulo π as X tends to infinity, and so from (3.1),

rq + 1 − a B(C/ ) = 2 , Fq 4q − a2 s ! 1 1 4q − a2 d P
= 1 − arccos . C/Fq;µ π 2 q + 1 − a

The Mertens conjecture for C/Fq therefore holds precisely when

rq + 1 − a 2 ≤ 1. 4q − a2

Upon squaring both sides and simplifying, we arrive at the inequality

(a − 2)2 ≤ 0, 32 Examples in Low Genus

which has only the solution a = 2, provided p 6= 2. Similarly, d PC/Fq;µ = 1 if and only if s ! 1 4q − a2 arccos = 0, 2 q + 1 − a which again holds only when a = 2 and p 6= 2. √ (2) If q = pm with a = ±2 q, where m is even, then from (3.6),

B(C/Fq) = ∞,
d PC/Fq;µ = 0.

X/2 For conditions (3)—(5), we find that MC/Fq (X)/q takes only finitely many X/2 values, so that the limiting distribution of MC/Fq (X)/q is simply a finite sum
of point masses. The natural density d PC/Fq;µ in each case is therefore given X/2 by the proportion of values taken by MC/Fq (X)/q that lie between −1 and 1. √ (3) (i) If q = pm with a = q, where m is even and p 6≡ 1 (mod 3), we have from (3.4) that   √     MC/ (X) πX 3 2 πX Fq = cos − 1 − √ sin . qX/2 3 3 q 3 We calculate the 6 cases of X (mod 6):

X/2 X (mod 6) MC/Fq (X)/q 0 1 √ 1 1/ q √ 2 −1 + 1/ q 3 −1 √ 4 −1/ q √ 5 1 − 1/ q

So

B(C/Fq) = 1,
d PC/Fq;µ = 1. √ (3) (ii) Similarly, if q = pm with a = − q, where m is even and p 6≡ 1 (mod 3),   √     MC/ (X) 2πX 3 2 2πX Fq = cos + 1 + √ sin . qX/2 3 3 q 3 The 3 cases of X (mod 3) are 3.2 Proof of Theorem 1.7 33

X/2 X (mod 3) MC/Fq (X)/q 0 1 √ 1 1/ q √ 2 −1 − 1/ q

This shows that 1 B(C/ ) = 1 + √ , Fq q 2 d P
= . C/Fq;µ 3 √ (4) (i) If q = 2m with a = 2q, where m is odd, then M (X) πX   1  πX  C/F2m = cos − 1 − sin . 2mX/2 4 2(m−1)/2 4 We analyse the 8 cases of X (mod 8):

mX/2 X (mod 8) MC/F2m (X)/2 0 1 1 2−m/2 2 −1 + 2−(m−1)/2 √ 3 − 2 + 2−m/2 4 −1 5 −2−m/2 6 1 − 2−(m−1)/2 √ 7 2 − 2−m/2

So  1 if m = 1, B(C/F2m ) = √ 1  2 − if m ≥ 3, 2m/2 
1 if m = 1, d PC/F2m ;µ = 3/4 if m ≥ 3. √ (4) (ii) Likewise, if q = 2m with a = − 2q, where m is odd, then M (X) 3πX   1  3πX  C/F2m = cos + 1 + sin . 2mX/2 4 2(m−1)/2 4 The table of values of X (mod 8) is 34 Examples in Low Genus

mX/2 X (mod 8) MC/F2m (X)/2 0 1 1 2−m/2 2 1 + 2−(m−1)/2 √ 3 2 + 2−m/2 4 −1 5 −2−m/2 6 −1 − 2−(m−1)/2 √ 7 − 2 − 2−m/2

Thus √ 1 B(C/ m ) = 2 + , F2 2m/2 1 d P
= . C/F2m ;µ 2 √ (4) (iii) If q = 3m with a = 3q, where m is odd, then M (X) πX   2  πX  C/F3m = cos − 1 − sin . 3mX/2 6 3m/2 6 The 12 cases of X (mod 12) are

mX/2 X (mod 12) MC/F3m (X)/3 0 1 √ 1 ( 3 − 1)/2 + 3−m/2 √ 2 −( 3 − 1)/2 + 3−(m−1)/2 3 −1 + 2 × 3−m/2 √ 4 −( 3 + 1)/2 − 3−(m−1)/2 √ 5 −( 3 + 1)/2 + 3−m/2 6 −1 √ 7 −( 3 − 1)/2 − 3−m/2 √ 8 ( 3 − 1)/2 − 3−(m−1)/2 9 1 − 2 × 3−m/2 √ 10 ( 3 + 1)/2 + 3−(m−1)/2 √ 11 ( 3 + 1)/2 − 3−m/2

Consequently, √ 3 + 1 1 B(C/ m ) = + , F3 2 3(m−1)/2 2 d P
= . C/F3m ;µ 3 3.2 Proof of Theorem 1.7 35

√ (4) (iv) Next, if q = 3m with a = − 3q, where m is odd, then

M (X) 5πX   2  5πX  C/F3m = cos + 1 + sin . 3mX/2 6 3m/2 6

Now

mX/2 X (mod 12) MC/F3m (X)/3 0 1 √ 1 −( 3 − 1)/2 + 3−m/2 √ 2 −( 3 − 1)/2 − 3−(m−1)/2 3 1 + 2 × 3−m/2 √ 4 −( 3 + 1)/2 − 3−(m−1)/2 √ 5 ( 3 + 1)/2 − 3−m/2 6 −1 √ 7 ( 3 − 1)/2 − 3−m/2 √ 8 ( 3 − 1)/2 + 3−(m−1)/2 9 −1 − 2 × 3−m/2 √ 10 ( 3 + 1)/2 + 3−(m−1)/2 √ 11 −( 3 + 1)/2 + 3−m/2

So we have that √ 3 + 1 1 B(C/ m ) = + , F3 2 3(m−1)/2 1 d P
= . C/F3m ;µ 2

(5) Finally, if q = pm with a = 0, where either m is even and p 6≡ 1 (mod 4), or m is odd, then     MC/ (X) πX 1 πX Fq = cos + √ sin . qX/2 2 q 2

The 4 cases of X (mod 4) are

X/2 X (mod 4) MC/Fq (X)/q 0 1 √ 1 1/ q 2 −1 √ 3 −1/ q 36 Examples in Low Genus

Thus

B(C/Fq) = 1,
d PC/Fq;µ = 1. Chapter 4

Global Mertens Conjectures

4.1 Averages over Families of Curves

In this section, we find a matrix theoretic expression for the average proportion of curves in a certain family for which the Mertens conjecture is true as the finite

field Fq grows larger. This involves expressing the quantity

MC/Fq (X) B(C/Fq) = lim sup X/2 X→∞ q in the language of unitary symplectic matrices. The space of unitary symplectic matrices USp2g(C) consists of 2g × 2g matrices U with complex entries satisfying   U †U = I and U T JU = J, where J = 0 Ig . The eigenvalues of U lie on −Ig 0 the unit circle and come in complex conjugate pairs, so that we may order the

iθ1 iθ2g eigenvalues e , . . . , e such that θj+g = −θj with 0 ≤ θj ≤ π for 1 ≤ j ≤ g. g Conversely, given (θ1, . . . , θg) ∈ [0, π] , the diagonal matrix with diagonal entries iθ1 iθg −iθ1 −iθg e , . . . , e , e , . . . , e lies in USp2g(C). Thus the set of conjugacy classes # g USp2g(C) of USp2g(C) corresponds to [0, π] .

Definition 4.1. For each U ∈ USp2g(C), we define the characteristic polynomial ZU (θ) for real θ by −iθ
ZU (θ) = det I − Ue .

Equivalently,

2g g Y i(θj −θ)
g Y iθ ZU (θ) = 1 − e = 2 e (cos θ − cos θj). (4.1) j=1 j=1

37 38 Global Mertens Conjectures

For a nonsingular projective curve C over Fq of genus g ≥ 1, there exists a # conjugacy class ϑ(C/Fq) in USp2g(C) , called the unitarised Frobenius conjugacy class attached to C/Fq, satisfying

2g e−iθ  Y Z (θ) = P √ = 1 − ei(θ(γj )−θ)
. (4.2) ϑ(C/Fq) C/Fq q j=1

That is, the eigenangles (θ1, . . . , θg) corresponding to the unitarised Frobenius conjugacy class ϑ(C/Fq) are precisely (θ(γ1), . . . , θ(γg)).

We shall find an expression for B(C/Fq) in terms of Zϑ(C/Fq)(θ) in the large q limit. For U ∈ USp2g(C), we define the function ϕ(U) by

2g X 1 ϕ(U) = , |Z 0(θ )| j=1 U j where eiθ1 , . . . , eiθ2g are the eigenvalues of U. We observe that ϕ depends only on the conjugacy class (θ1, . . . , θg) of U, and that ϕ is always nonnegative, though it blows up if U has a repeated eigenvalue. Note, however, that the set of matrices in USp2g(C) with repeated eigenvalues has measure zero with respect to the normalised Haar measure on USp2g(C). Lemma 4.2 (Cha [5, Equation (26)]). Suppose that C satisfies LI. Then we have that  1  B(C/ ) = ϕ (ϑ(C/ )) + O √ ϕ (ϑ(C/ )) Fq Fq g q Fq in the large q limit.

Proof. As C satisfies LI, we have from Theorem 2.6 that

X 1 γ B(C/ q) = . F Z 0 (γ−1) γ − 1 γ C/Fq Now by (2.1), 1 γ 1 − γ 0 −1 = 0 −1 , ZC/Fq (γ ) γ − 1 PC/Fq (γ ) whereas differentiating (4.2) shows that ie−iθ e−iθ  Z 0(θ) = − √ P 0 √ , (4.3) ϑ(C/Fq) q C/Fq q √ and so by taking θ = θ(γ), so that e−iθ(γ)/ q = γ−1, we find that

1 γ ie−2iθ(γ) 1 ie−iθ(γ) 0 −1 = 0 − √ 0 . ZC/Fq (γ ) γ − 1 Zϑ(C/Fq) (θ(γ)) q Zϑ(C/Fq) (θ(γ)) 4.1 Averages over Families of Curves 39

This yields the asymptotic   1 γ 1 1 1 0 −1 = 0 + Og √ 0 , ZC/Fq (γ ) γ − 1 |Zϑ(C/Fq) (θ(γ))| q |Zϑ(C/Fq) (θ(γ))| so by summing over all inverse zeroes γ, we obtain the desired identity.

We wish to determine the proportion of a family of curves that satisfy the Mertens conjecture. While we would like to choose this family to be as general as possible, it is imperative that we ensure that most curves in such a family satisfy LI, for otherwise it becomes significantly more difficult to analyse the behaviour of B(C/Fq). For this reason, we choose a family of hyperelliptic curves, as we shall show that we are then assured that LI holds for most such curves, with the added bonus of a framework for certain equidistribution results to hold. Via

Theorem 2.6, the former property yields yield a precise formula for B(C/Fq), while the latter allows us to use random matrix theory to compute averages in terms of integrals over USp2g(C). m Let q = p be a prime power with p > 2, n ≥ 1, and let Fqn be a finite field with qn elements. For g ≥ 1, let f be a monic polynomial of degree 2g + 1 with coefficients in Fqn whose discriminant is nonzero; equivalently, let f be a squarefree monic polynomials in Fqn [x] of degree 2g + 1. Each such polynomial f thereby defines a hyperelliptic curve Cf of genus g over Fqn via the affine model 2 y = f(x). So we let H2g+1,qn denote the set of these hyperelliptic curves C = Cf over Fqn . We are interested in properties of such curves C shared by “most” C ∈ H2g+1,qn .

Definition 4.3. We say that most hyperelliptic curves C ∈ H2g+1,qn , have the ∞ property D = {Dn}n=1 as n tends to infinity if

# {C ∈ H n : C satisfies D } lim 2g+1,q n = 1. n→∞ #H2g+1,qn

Theorem 4.4 (Chavdarov [6], Kowalski [16]; see [5, Theorem 3.1]). For fixed q and g ≥ 1, # {C ∈ H n : C satisfies LI} lim 2g+1,q = 1. n→∞ #H2g+1,qn

That is, as n tends to infinity, most hyperelliptic curves C ∈ H2g+1,qn , satisfy LI.

For brevity’s sake, we write C ∈ H2g+1,qn ∩ LI if C satisfies LI, and conversely c if C does not satisfy LI, we write C ∈ H2g+1,qn ∩ LI . 40 Global Mertens Conjectures

Proposition 4.5 (Deligne’s Equidistribution Theorem [13, Theorem 10.8.2]). Let f be a continuous function on USp2g(C) that is central, so that f is dependent only on the conjugacy class (θ1, . . . , θg) of each matrix U ∈ USp2g(C). Then for g ≥ 1,

1 X Z lim f (ϑ (C/Fqn )) = f(U) dµHaar(U), n→∞ #H2g+1,qn USp ( ) C∈H2g+1,qn 2g C where µHaar is the normalised Haar measure on USp2g(C).

Equivalently, consider the sequence of probability measures

1 X µn = δ ϑ(C/Fqn ) #H2g+1,qn C∈H2g+1,qn

# # on USp2g(C), where δU # is a point mass at a conjugacy class U ∈ USp2g(C) . Then Deligne’s equidistribution theorem merely states that the sequence of mea- sures µn converges weakly to µHaar as n tends to infinity. As USp2g(C) is a connected Lie group, and hence metrisable, we may apply the Portmanteau the- orem to the sequence of probability measures µn in order to obtain an equivalent reformulation of Deligne’s equidistribution theorem.

Corollary 4.6. For g ≥ 1,

# {C ∈ H2g+1,qn : ϑ (C/Fqn ) ∈ B} lim = µHaar(B) n→∞ #H2g+1,qn for any Borel set B ⊂ USp2g(C) whose boundary has Haar measure zero.

One can calculate this Haar measure precisely by using the following formula to convert it into an integral over [0, π]g.

Proposition 4.7 (Weyl Integration Formula [13, §5.0.4]). Let f be a bounded,

Borel-measurable complex-valued central function on USp2g(C). Then Z Z π Z π f(U) dµHaar(U) = ··· f(θ1, . . . , θg) dµUSp(θ1, . . . , θg), (4.4) USp2g(C) 0 0 where

g2 g 2 Y 2 Y dµ (θ , . . . , θ ) = (cos θ − cos θ ) sin2 θ dθ ··· dθ . (4.5) USp 1 g g!πg n m ` 1 g 1≤m

Lemma 4.8. Let B be an interval in R. Then the boundary of the set  U ∈ USp2g(C): ϕ(U) ∈ B has Haar measure zero.

Proof. By differentiating (4.1), we have that

g g 1 X Y 1 ϕ(U) = ϕ(θ1, . . . , θg) = g−1 cosec θj . (4.6) 2 |cos θk − cos θj| j=1 k=1 k6=j

So by the Weyl integration formula, we must show that for any interval B, the boundary of the set

g {(θ1, . . . , θg) ∈ [0, π] : ϕ(θ1, . . . , θg) ∈ B} has µUSp-measure zero. Observe that µUSp is absolutely continuous with respect to the Lebesgue measure on [0, π]g, and hence the sets

g {(θ1, . . . , θg) ∈ [0, π] : θj = θk for some 1 ≤ j < k ≤ g} , g {(θ1, . . . , θg) ∈ [0, π] : θj ∈ {0, π} for some 1 ≤ j ≤ g} have µUSp-measure zero; furthermore, the function ϕ is continuous on

 g (θ1, . . . , θg) ∈ [0, π] : 0 < θσ(1) < . . . < θσ(g) < π for each permutation σ of the set {1, . . . , g}. It therefore suffices to show that for each c ∈ R and for each permutation σ of {1, . . . , g}, the set

 g (θ1, . . . , θg) ∈ [0, π] : ϕ(θ1, . . . , θg) = c, 0 < θσ(1) < . . . < θσ(g) < π has µUSp-measure zero. But in the region where 0 < θσ(1) < . . . < θσ(g) < π, the function ϕ(θ1, . . . , θg) is not only continuous but real analytic and non-uniformly constant. As the zero set of a non-uniformly zero real analytic function has

Lebesgue measure zero, and µUSp is absolutely continuous with respect to the Lebesgue measure, we obtain the result.

Lemma 4.9. For all g ≥ 1, the function ϕ on USp2g(C) is integrable and satisfies the bounds Z 22g 0 ≤ ϕ(U) dµ (U) ≤ . Haar π USp2g(C) 42 Global Mertens Conjectures

This result follows from the following lemma in conjunction with the bound 2(g−1) 0 ≤ |ZU (θ)| ≤ 2 for all U ∈ USp2(g−1)(C) and θ ∈ [0, π], which is found by applying the triangle inequality to (4.1).

Lemma 4.10 (Cha [5, §4]). For g = 1, we have that

Z 4 ϕ(U) dµ (U) = , Haar π USp2(C) while for g ≥ 2, we have the identity

Z 2 Z π Z ϕ(U) dµ (U) = sin θ |Z (θ)| dµ (U) dθ. Haar π U Haar USp2g(C) 0 USp2(g−1)(C)

Proof. This is proved by Cha in [5, §4]; we include the details of the proof for later comparison. The g = 1 case is trivial, as in this case ϕ(U) = cosec θ, and hence by the Weyl integration formula,

Z 2 Z π 4 ϕ(U) dµ (U) = sin θ dθ = . Haar π π USp2(C) 0

We note that, strictly speaking, we require ϕ(U) to be bounded to use the Weyl integration formula, but we may replace ϕ(U) by ϕT (U) = min{ϕ(U),T } and then apply the monotone convergence theorem to obtain the above identity. For g ≥ 2, the Weyl integration formula together with the expression (4.6) for ϕ(U) gives

2g Z X 1 0 dµHaar(U) |ZU (θj)| USp2g(C) j=1   2 g g 2g Z π Z π 1 1  X Y  = g ··· g−1 cosec θj g!π 2 |cos θk − cos θj| 0 0 j=1 k=1 k6=j g Y 2 Y 2 × (cos θn − cos θm) sin θ` dθ1 ··· dθg. 1≤m

Now the expression in the brackets above is symmetric in the θn variables, so the summation on j may be replaced by g times a single summand, which we take to 4.1 Averages over Families of Curves 43

be θg. This allows us to rewrite the right-hand side as

2 g−1 2g −g+1 Z π Z π Y 1 g g ··· cosec θg g!π |cos θk − cos θg| 0 0 k=1 g Y 2 Y 2 × (cos θn − cos θm) sin θ` dθ1 ··· dθg 1≤m

From (4.1), we have that

g−1 g−1 Y 2 |cos θk − cos θg| = |ZU (θg)| , k=1 where U is an element of USp2(g−1)(C) in the conjugacy class (θ1, . . . , θg−1). We therefore have by the Weyl integration formula that Z 2 Z π Z ϕ(U) dµ (U) = sin θ |Z (θ )| dµ (U) dθ . Haar π g U g Haar g USp2g(C) 0 USp2(g−1)(C) We have now developed the necessary machinery needed in order to study the limit as n tends to infinity of the average

# {C ∈ H n : C satisfies the Mertens conjecture} 2g+1,q , #H2g+1,qn which may be thought of as a geometric average of the number of hyperelliptic curves in H2g+1,qn satisfying the Mertens conjecture. For brevity’s sake, we write this average as # {C ∈ H n ∩ Mertens} 2g+1,q . #H2g+1,qn Proposition 4.11. We have that

# {C ∈ H2g+1,qn ∩ Mertens} 
lim = µHaar U ∈ USp2g(C): ϕ(U) ≤ 1 . n→∞ #H2g+1,qn Proof. For any ε > 0, we may write

# {C ∈ H2g+1,qn ∩ Mertens} = A1 + A2 + A3 + A4 + A5 + A6 + A7, 44 Global Mertens Conjectures

where

A1 = # {C ∈ H2g+1,qn : ϕ (ϑ (C/Fqn )) ≤ 1} , c A2 = −# {C ∈ H2g+1,qn ∩ LI : ϕ (ϑ (C/Fqn )) ≤ 1} , c A3 = # {C ∈ H2g+1,qn ∩ Mertens ∩ LI } ,

A4 = # {C ∈ H2g+1,qn ∩ LI : B (C/Fqn ) ≤ 1, 1 < ϕ (ϑ (C/Fqn )) ≤ 1 + ε} ,

A5 = −# {C ∈ H2g+1,qn ∩ LI : B (C/Fqn ) > 1, 1 − ε ≤ ϕ (ϑ (C/Fqn )) ≤ 1} ,

A6 = # {C ∈ H2g+1,qn ∩ LI : B (C/Fqn ) ≤ 1, ϕ (ϑ (C/Fqn )) > 1 + ε} ,

A7 = −# {C ∈ H2g+1,qn ∩ LI : B (C/Fqn ) > 1, ϕ (ϑ (C/Fqn )) < 1 − ε} ,

By Deligne’s equidistribution theorem,

A1 lim = µHaar(ϕ(U) ≤ 1), n→∞ #H2g+1,qn while Theorem 4.4 implies that A A lim 2 = lim 3 = 0. n→∞ #H2g+1,qn n→∞ #H2g+1,qn Next, we note that

|A4| + |A5| ≤ # {C ∈ H2g+1,qn : 1 − ε ≤ ϕ (ϑ (C/Fqn )) ≤ 1 + ε} , and hence |A4| + |A5| lim sup ≤ µHaar(1 − ε ≤ ϕ(U) ≤ 1 + ε). n→∞ #H2g+1,qn by Deligne’s equidistribution theorem. Finally, Lemma 4.2 implies the existence of a constant c(g) > 0 such that

 n/2 |A6| + |A7| ≤ # C ∈ H2g+1,qn : ϕ (ϑ (C/Fqn )) ≥ εc(g)q . Lemma 4.10 shows that ϕ(U) is integrable, which implies that for any ε0 > 0 0 there exists some T0 > 0 such that µHaar (ϕ(U) ≥ T ) ≤ ε for all T ≥ T0. Thus for any ε0 > 0, we have by Deligne’s equidistribution theorem that

 n/2 |A6| + |A7| # C ∈ H2g+1,qn : ϕ (ϑ (C/Fqn )) ≥ εc(g)q lim sup ≤ lim sup n→∞ #H2g+1,qn n→∞ #H2g+1,qn

# {C ∈ H n : ϕ (ϑ (C/ n )) ≥ T } ≤ lim 2g+1,q Fq n→∞ #H2g+1,qn

= µHaar (ϕ(U) ≥ T ) ≤ ε0. 4.1 Averages over Families of Curves 45

As ε0 > 0 was arbitrary, A A lim 6 = lim 7 = 0. n→∞ #H2g+1,qn n→∞ #H2g+1,qn So we have shown that for any ε > 0,

# {C ∈ H2g+1,qn ∩ Mertens} lim − µHaar (ϕ(U) ≤ 1) n→∞ #H2g+1,qn

≤ µHaar (1 − ε ≤ ϕ(U) ≤ 1 + ε) .

As ε > 0 was arbitrary, and

lim µHaar (1 − ε ≤ ϕ(U) ≤ 1 + ε) = µHaar (ϕ(U) = 1) = 0, ε→0 we obtain the result.

So in order to prove Theorem 1.8, we must show that µHaar (ϕ(U) ≤ 1) = 0 for 1 ≤ g ≤ 2. Here the minimum of ϕ can be determined explicitly, and in particular it can be shown that the set

g {(θ1, . . . , θg) ∈ [0, π] : ϕ (θ1, . . . , θg) ≤ 1} is finite; this then implies the result via the Weyl integration formula, together with the fact that the measure µUSp is atomless, with µUSp as in (4.5). More precisely, for each permutation σ of the set {1, . . . , g}, there is precisely one global minimum of ϕ in the region

 g (θ1, . . . , θg) ∈ [0, π] : 0 < θσ(1) < . . . < θσ(g) < π ,   with this minimum occurring at the critical point θeσ(1),..., θeσ(g) , where

   π 3π (2g − 1)π  θe1,..., θeg = , ,..., . (4.7) 2g 2g 2g

One can interpret this result via a geometric argument. If z1, . . . , zg are g points on the unit circle in the complex plane, then we may consider the product of the chord lengths of chords from a single point zj to the other g − 1 points and also to the complex conjugate of zj. We can then think of ϕ as the sum over the inverse of this product for each starting point zj. Intuitively, we would expect the product of chord lengths to be largest when averaged over the starting points when the g-tuple of points on the unit circle are evenly spaced while simultaneously being as far as possible from the points ±1; consequently, we would expect ϕ to be smallest at this same g-tuple. 46 Global Mertens Conjectures

Proof of Theorem 1.8. For g = 1, we have from (4.6) that

ϕ (θ1) = cosec θ1, which is always at least 1, and is exactly 1 only at the point θ1 = π/2. For g = 2, 1 1 ϕ (θ1, θ2) = (cosec θ1 + cosec θ2) 2 |cos θ1 − cos θ2| so for this to be at most 1, we must have that

f (θ1, θ2) = 2 |cos θ1 − cos θ2| − cosec θ1 − cosec θ2 ≥ 0.

Now when 0 < θ1 < θ2 < π, we have that ∂f = −2 sin θ1 + cosec θ1 cot θ1, ∂θ1 ∂f = 2 sin θ2 + cosec θ2 cot θ2. ∂θ2

3 We set both of these to zero, multiply through by sin θ (with θ = θ1 for the former and θ2 for the latter), subtract cos θ, and then square both sides, finding that in both cases,

4x6 + x2 − 1 = 2x2 − 1
2x4 + x2 + 1
= 0, √ where x = sin θ. As 0 < θ < π, this has only the solution x = 1/ 2, or equivalently θ ∈ {π/4, 3π/4}. So the only critical point of f in the region 0 <

θ1 < θ2 < π occurs at (θ1, θ2) = (π/4, 3π/4); we may easily confirm that ∂f/∂θ1 =

∂f/∂θ2 = 0 at this point, and also determine that f (π/4, 3π/4) = 0. So it remains to show that this is a local maximum of f. Indeed,

2 ∂ f 2 3 2 = −2 cos θ1 − cosec θ1 cot θ1 − cosec θ1, ∂θ1 ∂f 2 3 = 2 cos θ2 − cosec θ2 cot θ2 − cosec θ2, ∂θ2 ∂2f ∂2f = = 0, ∂θ1∂θ2 ∂θ2∂θ1 and in particular, the Hessian matrix of second partial derivatives of f evaluated at (π/4, 3π/4) is √ ! −4 2 0 √ , 0 −2 2 4.2 The Critical Point 47

which is negative definite. So (π/4, 3π/4) is a local maximum of f, and as f tends to negative infinity as either θ1 or θ2 tends to 0 or π, and f (θ1, θ1) = −2 cosec θ1 < 0, the point (π/4, 3π/4) is the unique global maximum of f on the set where

0 < θ1 < θ2 < π. The same argument shows that the unique global maximum of f when 0 < θ2 < θ1 < π occurs at the point (3π/4, π/4), with f (3π/4, π/4) = 0.

Consequently, these are the only two points where ϕ (θ1, θ2) ≤ 1. While we we can prove this result for 1 ≤ g ≤ 2, we are in fact able to show that the critical point (4.7) is a local minimum of ϕ for every positive integer g; however, we are yet to be able to prove that this is also a global minimum. The proof is long, and so we dedicate the entirety of the next section to showing this result.

4.2 The Critical Point

We first demonstrate that ϕ(θ1, . . . , θg) = 1 at the critical point.   Lemma 4.12. Let θe1,..., θeg be as in (4.7). Then for each permutation σ of {1, . . . , g}, we have that   ϕ θeσ(1),..., θeσ(g) = 1. Proof. It suffices to prove this when σ is the identity, as ϕ is invariant under permutations of the variables. So by differentiating (4.1), we have that g g X 1 X ϕ (θ , . . . , θ ) = 2 = 2 ϕ (θ , . . . , θ ), 1 g |Z 0(θ )| j 1 g j=1 U j j=1 where for each 1 ≤ j ≤ g, g 1 Y 1 ϕj (θ1, . . . , θg) = |1 − e2iθj | 1 − ei(θj −θk) 1 − ei(θj +θk) k=1 k6=j g (4.8) 1 Y 1 = g cosec θj , 2 |cos θk − cos θj| k=1 k6=j so that g   1 Y 1 ϕj θe1,..., θeg = |1 − e2πi(2j−1)/(2g)| |1 − e2πi(j−k)/(2g)| |1 − e2πi(j+k−1)/(2g)| k=1 k6=j 2g−1 Y 1 = , |1 − e2πik/(2g)| k=1 48 Global Mertens Conjectures

as for each j, the set {2j −1, j −k, j +k −1 : 1 ≤ k ≤ g, k 6= j} forms a complete set of residues modulo 2g but for an element 0 modulo 2g. Taking x = 1 in the identity 2g−1 2g−1 X Y xj = x − e2πik/(2g)
, j=0 k=1 we find that for each 1 ≤ j ≤ g,   1 ϕj θe1,..., θeg = , (4.9) 2g which yields the result.   To confirm that the point θe1,..., θeg is indeed correctly identified as a critical   point, we must next show that the derivative of ϕ vanishes at θe1,..., θeg .   Lemma 4.13. Let θe1,..., θeg be as in (4.7). Then for each permutation σ of   {1, . . . , g}, the derivative of ϕ vanishes at the point θeσ(1),..., θeσ(g) . Proof. Again, we need only prove this when σ is the identity. Upon differentiating (4.8), we have that for 1 ≤ j, m ≤ g with j 6= m,

∂ϕj sin θm = ϕj ∂θm cos θm − cos θj 1  θ − θ  θ + θ  = − cot m j + cot m j ϕ , (4.10) 2 2 2 j with ϕj = ϕj (θ1, . . . , θg) as in (4.8); here the second equality follows from the sine angle-sum and cosine sum-to-product formulæ. If j = m, then g      ∂ϕm 1 X θm − θj θm + θj = − cot + cot ϕ − cot θ ϕ . (4.11) ∂θ 2 2 2 m m m m j=1 j6=m

Letting θk = θek = (2k − 1)π/(2g) for 1 ≤ k ≤ g and using (4.9), we find that for j 6= m,      ∂ϕj   1 (m − j)π (m + j − 1)π θe1,..., θeg = − cot + cot , (4.12) ∂θm 4g 2g 2g while when j = m,

∂ϕm   θe1,..., θeg ∂θm g 1 X  (m − j)π  (m + j − 1)π  1 (2m − 1)π  = − cot + cot − cot . 4g 2g 2g 2g 2g j=1 j6=m 4.2 The Critical Point 49

Now the cotangent function has period π and is odd about the origin, and con- sequently

g 2g−1 X  (m − j)π  (m + j − 1)π  X jπ  cot + cot = cot 2g 2g 2g j=1 j=1 j6=m j6=2m−1 (4.13) (2m − 1)π  = − cot . 2g

So   ∂ϕm   1 (2m − 1)π θe1,..., θeg = − cot , (4.14) ∂θm 4g 2g while g   X ∂ϕj   1 (2m − 1)π θe1,..., θeg = cot . ∂θ 4g 2g j=1 m j6=m We therefore find that for each 1 ≤ m ≤ g,

g ∂ϕ   X ∂ϕj   θe1,..., θeg = 2 θe1,..., θeg = 0, ∂θ ∂θ m j=1 m as required.   Next, we show that θe1,..., θeg is a local minimum of ϕ, by calculating the   Hessian matrix of ϕ at θe1,..., θeg and proving it to be positive definite.   Lemma 4.14. Let θe1,..., θeg be as in (4.7). Then for each permutation σ of   {1, . . . , g}, the point θeσ(1),..., θeσ(g) is a local minimum of ϕ.

Proof. Again, we need only take σ to be the identity. We first determine the mixed partial derivatives of ϕ. When 1 ≤ j, m, n ≤ g with j, m, n distinct, we differentiate (4.10) to find that

∂2ϕ ∂2ϕ 1  θ − θ  θ + θ  ∂ϕ j = j = − cot m j + cot m j j , ∂θn∂θm ∂θm∂θn 2 2 2 ∂θn while when m 6= n, differentiating (4.11) yields

∂2ϕ ∂2ϕ 1  θ − θ  θ + θ  ∂ϕ m = m = − cot n m + cot n m m ∂θn∂θm ∂θm∂θn 2 2 2 ∂θm 1  θ − θ  θ + θ  − cosec2 n m − cosec2 n m ϕ . 4 2 2 m 50 Global Mertens Conjectures

So taking θk = θek = (2k − 1)π/(2g) for 1 ≤ k ≤ g, we have by (4.12) that

2      ∂ ϕj   1 (m − j)π (m + j − 1)π θe1,..., θeg = cot + cot ∂θn∂θm 8g 2g 2g  (n − j)π  (n + j − 1)π  × cot + cot . 2g 2g

By expanding this product and using the cosine and sine angle-sum formulæ on each term, as well as the fact that the cotangent function is odd about the origin, we find that this is identical to

1  (m − n)π  (m + n − 1)π  cot + cot 8g 2g 2g  (n − j)π  (n + j − 1)π  × cot + cot 2g 2g 1  (n − m)π  (n + m − 1)π  + cot + cot 8g 2g 2g  (m − j)π  (m + j − 1)π  × cot + cot . 2g 2g

Also, (4.14) and (4.9) show that

2 ∂ ϕm   θe1,..., θeg ∂θn∂θm 1  (n − m)π  (n + m − 1)π  (2m − 1)π  = cot + cot cot 8g 2g 2g 2g 1  (n − m)π  (n + m − 1)π  − cosec2 − cosec2 . 8g 2g 2g

We therefore find that for 1 ≤ m, n ≤ g with m 6= n,

2 g 2 ∂ ϕ   X ∂ ϕj   θe1,..., θeg = 2 θe1,..., θeg ∂θ ∂θ ∂θ ∂θ n m j=1 n m 1  (n − m)π  (n + m − 1)π  = cot2 − cot2 2g 2g 2g 1  (n − m)π  (n + m − 1)π  − cosec2 − cosec2 2g 2g 2g = 0, where we have used (4.13), the fact that the cotangent function is odd about the origin, and the Pythagorean trigonometric identity cosec2 θ − cot2 θ = 1. 4.2 The Critical Point 51

When 1 ≤ j, m ≤ g with j 6= m, we also have by differentiating (4.10) that

2      ∂ ϕj 1 θm − θj θm + θj ∂ϕj 2 = − cot + cot ∂θm 2 2 2 ∂θm 1  θ − θ  θ + θ  + cosec2 m j + cosec2 m j ϕ , 4 2 2 j while differentiating (4.11) shows that when j = m,

2 g      ∂ ϕm 1 X θm − θj θm + θj ∂ϕm = − cot + cot ∂θ2 2 2 2 ∂θ m j=1 m j6=m g      1 X θm − θj θm + θj + cosec2 + cosec2 ϕ 4 2 2 m j=1 j6=m

∂ϕm 2 − cot θm + cosec θmϕm. ∂θm

So when θk = θek = (2k − 1)π/(2g) for 1 ≤ k ≤ g, we have by (4.12) and (4.9) that 2     2 ∂ ϕj   1 (m − j)π (m + j − 1)π 2 θe1,..., θeg = cot + cot ∂θm 8g 2g 2g 1  (m − j)π  (m + j − 1)π  + cosec2 + cosec2 , 8g 2g 2g while (4.14), (4.13), and (4.9) imply that

2 g      ∂ ϕm   1 X 2 (m − j)π 2 (m + j − 1)π θe1,..., θeg = cosec + cosec ∂θ2 8g 2g 2g m j=1 j6=m 1 (2m − 1)π  1 (2m − 1)π  + cot2 + cosec2 . 8g 2g 2g 2g Thus 2 g 2 ∂ ϕ   X ∂ ϕj   θe1,..., θeg = 2 θe1,..., θeg ∂θ2 ∂θ2 m j=1 m 2g−1 2g−1 1 X jπ  1 X jπ  = cot2 + cosec2 4g 2g 2g 2g j=1 j=1 g 1 X (m − j)π  (m + j − 1)π  + cot cot 2g 2g 2g j=1 j6=m 1 (2m − 1)π  + cosec2 , 2g 2g 52 Global Mertens Conjectures

where we have used the fact that cot2 θ and cosec2 θ have period π. Now using the cosine and sine angle-sum formulæ, as well as (4.13), we determine that g 1 X (m − j)π  (m + j − 1)π  1 1 1 (2m − 1)π  cot cot = − − cot2 , 2g 2g 2g 2 2g 2g 2g j=1 j6=m so our previous calculations simplify to

2 2g−1   2g−1   ∂ ϕ   1 X 2 jπ 1 X 2 jπ 1 θe1,..., θeg = cot + cosec + , ∂θ2 4g 2g 2g 2g 2 m j=1 j=1 where we have once again used the Pythagorean trigonometric identity. So we have shown that the Hessian matrix of second partial derivatives of   ϕ evaluated at θe1,..., θeg is diagonal, and furthermore each diagonal entry is   strictly positive. Thus this matrix is positive definite, and so θe1,..., θeg is a local minimum of ϕ.

This result does not preclude the possibility of the existence of other, possible smaller, local minima of ϕ. Brendan Harding (personal communication) has performed numerical calculations for small values of g to find other possible local minima of ϕ. Via the gradient descent method, he has searched for local minima of ϕ for each 1 ≤ g ≤ 50; his results have so far only indicated the existence of   a local minimum at the critical point θe1,..., θeg as in (4.7). Nevertheless, this does not eliminate the possibility of other such local minima, though it does seem extremely unlikely. We must also mention that despite these results being formulated only for unitary symplectic matrices, they can easily be extended to hold for unitary matrices. Indeed, if U is an N × N unitary matrix, so that U has eigenvalues

iθ1 iθN e , . . . , e with −π ≤ θj ≤ π for all 1 ≤ j ≤ N, then for real θ,

N −iθ
Y i(θj −θ)
ZU (θ) = det I − Ue = 1 − e , j=1 so that N N N X 1 X Y 1 ϕ(U) = = . 0 i(θ −θ ) |ZU (θj)| 1 − e k j j=1 j=1 k=1 k6=j Then the same methods as in Lemmata 4.12, 4.13, and 4.14 show that any per- mutation σ and any one-dimensional translation φ modulo 2π of the critical point    (N − 1)π (N − 3)π (N − 1)π  θe1,..., θeN = − , − ,..., N N N 4.3 Variants of the Mertens Conjecture 53

is a local minimum of ϕ, with   ϕ θeσ(1) + φ, . . . , θeσ(N) + φ = 1.

Furthermore, we are led to conjecture that these points are precisely the global minima of ϕ. We recover our conjecture for unitary symplectic matrices by letting

N = 2g and restricting ourselves to the subgroup of matrices for which θj+g = −θj with 0 ≤ θj ≤ π for 1 ≤ j ≤ g.

4.3 Variants of the Mertens Conjecture

It is worth noting that there are variants of MC/Fq (X) that can be studied. One can consider certain weights involved in the summatory function of the M¨obius function. In the classical case, we may instead look at the properties of the weighted sum X µ(n) M (x) = α nα n≤x for α ∈ R. The function field analogue is X−1 X 1 X M (X) = µ (D), C/Fq,α qα(N+1) C/Fq N=0 deg(D)=N and we may well ask whether the α-Mertens conjecture,

MC/Fq,α(X) lim sup (1/2−α)X ≤ 1, (4.15) X→∞ q holds for the function field C/Fq. For α > 1, this is easily resolved; (2.3) and

(2.4) show that MC/Fq,α(X) converges to the infinite series ∞ 1 X 1 X 1 α αN µC/Fq (D) = α −α . (4.16) q q q ZC/ (q ) N=0 deg(D)=N Fq Though this series is only absolutely convergent for |u| < q−1, one can show that it is also conditionally convergent for |u| < q−1/2 due to the lack of poles of

1/ZC/Fq (u) inside this disc, and so MC/Fq,α(X) also converges to the quantity in (4.16) for 1/2 < α ≤ 1. For α < 1/2, on the other hand, a minor modification of the proof of Proposition 2.4, essentially involving dividing (2.9) by qα(N+1) and summing over all 0 ≤ N ≤ X − 1, shows that   MC/ q,α(X) X 1 γ 1 F = − eiθ(γ)X + O q(1/2−α)X Z 0 (γ−1) γ − qα q,g q(1/2−α)X γ C/Fq 54 Global Mertens Conjectures

provided ZC/Fq (u) has only simple zeroes. A similarly simple modification of the proof of Lemma 4.2 then shows that if C satisfies LI, the quantity

MC/Fq,α(X) Bα(C/Fq) = lim sup (1/2−α)X X→∞ q satisfies the asymptotic  1  B (C/ ) = ϕ (ϑ(C/ )) + O ϕ (ϑ(C/ )) α Fq Fq g,α q1/2−α Fq as q tends to infinity. So for α < 1/2, the proof of Theorem 1.8 can be adapted essentially unchanged with the Mertens conjecture for the function field C/Fq re- placed by (4.15). Thus for α < 1/2, while any formulation of a local Mertens con- jecture for MC/Fq,α(X) may differ to those involving MC/Fq (X), a global Mertens conjecture would not. For α = 1/2, it is more prudent to analyse the the properties of the M¨obius function more locally by merely studying the behaviour of 1 X µ (D) q(N+1)/2 C/Fq deg(D)=N for each N ≥ 0, rather than MC/Fq,1/2(X), its average over 0 ≤ N ≤ X −1. While this is not useful in the classical case, where this would simply be ascertaining µ(n) for each n ≥ 1, the function field case is quite practical. From the proof of Proposition 2.4,   1 X X 1 iθ(γ)(N+1) 1 (N+1)/2 µC/Fq (D) = − 0 −1 e + Oq,g N/2 , q ZC/ (γ ) q deg(D)=N γ Fq and then a minor modification of the proof of Lemma 4.2 shows that the quantity

1 X B1/2(C/Fq) = lim sup (N+1)/2 µC/Fq (D) N→∞ q deg(D)=N satisfies  1  B (C/ ) = ϕ (ϑ(C/ )) + O √ ϕ (ϑ(C/ )) 1/2 Fq Fq g q Fq as q tends to infinity, just as B(C/Fq) does so, and hence that Theorem 1.8 re- mains true with the Mertens conjecture for the function field C/Fq being replaced by the modified Mertens conjecture

1 X lim sup (N+1)/2 µC/Fq (D) ≤ 1. N→∞ q deg(D)=N 4.3 Variants of the Mertens Conjecture 55

A further variant follows from noting that while the classical Mertens conjec- ture states that the inequality |M(x)| √ ≤ 1 (4.17) x holds for all x ≥ 1, the value 1 on the right-hand side above is, in some sense, not particularly special. Indeed, Stieltjes [27] claimed to have a proof that √ M(x) = O x
without specifying an explicit constant, before later rescinding his claim, though he did postulate that (4.17) was true. Similarly, von Sterneck [26] conjectured that the stronger inequality |M(x)| 1 √ ≤ x 2 holds for all x ≥ 200, based on calculations of M(x) up to 5 000 000. In spite of these na¨ıve conjectures, however, it seems most likely that M(x) lim sup √ = ∞, x→∞ x M(x) lim inf √ = −∞. x→∞ x It is not difficult to prove this to be true should the Riemann hypothesis prove to be false, while Ingham [12] showed that this result also follows if one as- sumes the Riemann hypothesis and the Linear Independence hypothesis for the Riemann zeta function. Furthermore, the work of Ng [21] does not merely con- ditionally show that the set of counterexamples to the Mertens conjecture has strictly positive logarithmic density: the same can actually be said for the set √ {x ∈ [1, ∞): |M(x)| > β x} for any β > 0. One may very well then ask if the value 1 on the right-hand side of the Mertens conjecture in function fields,

MC/Fq (X) lim sup X/2 ≤ 1, X→∞ q is crucial in our analysis so far. We may instead consider the following general- isation of the Mertens conjecture in function fields: for β > 0, we say that the function field of a curve C over a finite field Fq satisfies the β-Mertens conjecture if MC/Fq (X) lim sup X/2 ≤ β. X→∞ q 56 Global Mertens Conjectures

We can then study the average

# {C ∈ H n : C satisfies the β-Mertens conjecture} 2g+1,q , #H2g+1,qn which we abbreviate to

# {C ∈ H n ∩ β-Mertens} 2g+1,q . #H2g+1,qn Theorem 4.15. If 0 < β ≤ 1, and if 1 ≤ g ≤ 2 is fixed, most hyperelliptic curves

C ∈ H2g+1,qn of genus g do not satisfy the β-Mertens conjecture for C/Fqn . If β > 1, then for any fixed g ≥ 1,

# {C ∈ H n ∩ β-Mertens} 0 < lim 2g+1,q < 1. (4.18) n→∞ #H2g+1,qn Proof. A simple modification of the proof of Proposition 4.11 shows that

# {C ∈ H2g+1,qn ∩ β-Mertens} 
lim = µHaar U ∈ USp2g(C): ϕ(U) ≤ β . n→∞ #H2g+1,qn It is clear that this is nondecreasing in β, and so the proof of Theorem 1.8 implies that this is equal to zero for 0 < β ≤ 1 and 1 ≤ g ≤ 2. To prove the inequalities

(4.18) for β > 1, we recall from Lemma 4.12 that the equality ϕ (θ1, . . . , θg) = 1 is attained in the region 0 < θ1 < . . . < θg < 1, and ϕ is real analytic and not uniformly constant in this region, and hence there exists an open neighbourhood of the point in this region where 1 ≤ ϕ (θ1, . . . , θg) ≤ β. This open neighbourhood must have positive µUSp-measure, as dµUSp (θ1, . . . , θg) does not vanish on open g subsets of [0, π] . Consequently, µHaar (ϕ(U) ≤ β) > 0. On the other hand, we must also have that µHaar (ϕ(U) ≤ β) < 1, as ϕ blows up when θj = θk for any j 6= k, and so for any such point there exists some open neighbourhood with

ϕ (θ1, . . . , θg) > β in this neighbourhood.

The situation in function fields is therefore markedly different to the classical case. The work of Ng shows that in the classical case, the set of “local” coun- √ terexamples x ∈ [1, ∞) to the β-Mertens conjecture |M(x)| ≤ β x has positive logarithmic density for all β > 0. In the function field case, where we instead “globally” consider the proportion of curves for which the β-Mertens conjecture is true, the value β = 1 truly is the optimal value of β, in the sense that it is the largest such β for which most hyperelliptic curves C ∈ H2g+1,qn , of genus 1 ≤ g ≤ 2 do not satisfy the β-Mertens conjecture. Chapter 5

P´olya’s Conjecture in Function Fields

5.1 P´olya’s Conjecture

The Liouville function λ(n) is the arithmetic function that counts, modulo 2, the number of prime numbers dividing a positive integer, counting multiplicity. That is,  1 if n = 1,  λ(n) = −1 if n has an odd number of prime factors, counting multiplicity,  1 if n has an even number of prime factors, counting multiplicity.

In particular, the Liouville function agrees with the M¨obiusfunction on the squarefree positive integers. In 1919, P´olya [23] conjectured that the summa- tory function of the Liouville function, X L(x) = λ(n), n≤x satisfies the inequality L(x) ≤ 0 (5.1) for all x ≥ 2; P´olya remarked in [23] that he had checked the validity of his conjecture up to x = 1500. Much like the Mertens conjecture, this conjecture implies that the Riemann hypothesis is true and that the Riemann zeta func- tion has only simple zeroes. P´olya’s conjecture also shared the same fate as the Mertens conjecture, in that it was proven to be false; it was disproved by Hasel- grove [9] in 1958 using methods closely related to the work of Ingham [12]. The

57 58 P´olya’s Conjecture in Function Fields

first counterexample was later shown to occur at x = 906 150 257 [28], and it is now known [3] that L(x) lim sup √ > 0.062, x→∞ x L(x) lim inf √ < −1.389. x→∞ x It seems likely that L(x) lim sup √ = ∞, x→∞ x L(x) lim inf √ = −∞, x→∞ x and this is known to follow from the assumption of the Linear Independence hypothesis for the Riemann zeta function [12]. In spite of these results, numerical evidence [3] suggests that regions for which the conjectured inequality (5.1) fails are distributed rather sparsely amongst the positive integers. Analogously to the Mertens conjecture, this can be explained heuristically through the following explicit expression for L(x) in terms of a sum over the nontrivial zeroes ρ of the Riemann zeta function.

Proposition 5.1 (Fawaz [8], Humphries [11, Theorem 4.5]; cf. Proposition 1.1). Assume the Riemann hypothesis and the simplicity of the zeroes of ζ(s). Then ∞ there exists a sequence {Tv}v=1 with v ≤ Tv ≤ v + 1 such that for each positive integer v, for all ε > 0, and for x a positive noninteger, √ x X ζ(2ρ) xρ  1 x log x x  √ L(x) = + 0 + 1 + Oε + + 1−ε . ζ(1/2) ζ (ρ) ρ x Tv Tv log x |γ|

L(x) 1 X ζ(2ρ) xiγ 1  1  √ = + + √ + O , (5.2) x ζ(1/2) ζ0(ρ) ρ x x ρ where the sum P is interpreted in the sense lim P . The leading term ρ v→∞ |γ|

P´olya’s conjecture holds. We can guarantee the existence of this logarithmic density, and more, if in addition to the Riemann hypothesis we assume the Linear Independence hypothesis as well as a certain conjecture on the growth rate of ζ0(ρ) P 0 −2 for each nontrivial zero ρ of ζ(s); that is, we assume that |γ|

Theorem 5.2 (Humphries [11, Theorem 5.1]; cf. [21], [25]). Assume the Riemann hypothesis, the Linear Independence hypothesis, and that P |ζ0(ρ)|−2  T . √ |γ|

The last point here yields an upper bound on the logarithmic density of Pλ, while a method of Montgomery [18] also yields a lower bound.

Corollary 5.3 (Humphries [11, Theorem 1.5]). Under the same assumptions as Theorem 5.2, we have the bounds

1/2 ≤ δ (Pλ) < 1.

Richard Brent (personal communication) has subsequently performed calcu- lations that suggest that the true value of this logarithmic density is

δ (Pλ) ≈ 0.99988,

In any case, we may say that conditionally L(x) does indeed have a bias towards being nonpositive, but that the set of counterexamples to P´olya’s conjecture is not insignificant, in that it has positive, although very small, logarithmic density.

5.2 P´olya Conjectures in Function Fields

Here we formulate several function field analogues of P´olya’s conjecture. We first define the Liouville function of a function field. For C a nonsingular projective 60 P´olya’s Conjecture in Function Fields

curve over Fq of genus g and D an effective divisor of C, the Liouville function of C/Fq is given by  1 if D is the zero divisor,  λC/Fq (D) = −1 if the sum of the orders of the prime divisors of D is odd,  1 if the sum of the orders of the prime divisors of D is even.

We then study the summatory function of the Liouville function of C/Fq,

X−1 X X LC/Fq (X) = λC/Fq (D). N=0 deg(D)=N

We wish to know whether there are biases in the behaviour of LC/Fq (X). Like the classical case, we find that LC/Fq (X) may have a bias towards being nonpositive.

P´olya’s Conjecture in Function Fields. Let C be a nonsingular projective curve over Fq of genus g, and let LC/Fq (X) be the summatory function of the Liouville function of C/Fq. Then

lim sup LC/Fq (X) ≤ 0. X→∞

As with the Mertens conjecture, we may consider both local and global ques- tions pertaining to P´olya’s conjecture in function fields.

Question 5.4. For which curves does P´olya’sconjecture hold?

Question 5.5. Given a function field of a curve C over a finite field Fq, how frequently does the inequality

LC/Fq (X) ≤ 0 (5.3) hold?

Question 5.6. On average, in either the q or the g aspect, how often does P´olya’s conjecture hold?

The local questions are addressed in Chapter 6. In Section 6.1, we formulate certain conditions on the zeroes of ZC/Fq (u) to ensure that P´olya’s conjecture for C/Fq is true. 5.2 P´olya Conjectures in Function Fields 61

Theorem 5.7 (cf. Theorem 1.5). Let C be a nonsingular projective curve over

Fq of genus g ≥ 1. If ZC/Fq (u) has only simple zeroes, then P´olya’sconjecture for C/Fq is true provided √ −g √ −g −2 1 q q hC/Fq 1 q q hC/Fq X ZC/Fq (γ ) γ − √ + √ ≤ − , 2 q + 1 P (q−1/2) 2 q − 1 P (−q−1/2) Z 0 (γ−1) γ − 1 C/Fq C/Fq γ C/Fq (5.4) where hC/Fq is the class number of the function field C/Fq. Furthermore, if C satisfies LI, then the converse is also true: P´olya’sconjecture for C/Fq is true only when (5.4) holds.

This does not entirely answer Question 5.4; it is possible that C/Fq is such that (5.4) does not hold but that P´olya’s conjecture for C/Fq is true; in order for this to happen, ZC/Fq (u) must only has simple zeroes but C must fail to satisfy LI. We deal with Question 5.5 in Section 6.3, where we determine the natural density of the set of positive integers X for which (5.3) holds, provided that C satisfies LI.

Theorem 5.8 (cf. Theorem 1.6). Let C be a nonsingular projective curve over

Fq of genus g ≥ 1, and suppose that C satisfies LI. The natural density 1  d(PC/ ;λ) = lim # 1 ≤ X ≤ Y : LC/ (X) ≤ 0 Fq Y →∞ Y Fq exists and satisfies

d(PC/Fq;λ) = 1/2 if −φ1(C/Fq) + φ2(C/Fq) ≥ φ3(C/Fq),

1/2 < d(PC/Fq;λ) < 1 if −φ3(C/Fq) < −φ1(C/Fq) + φ2(C/Fq) < φ3(C/Fq),

d(PC/Fq;λ) = 1 if −φ1(C/Fq) + φ2(C/Fq) ≤ −φ3(C/Fq), where √ −g 1 q q hC/Fq φ1(C/Fq) = √ −1/2 , 2 q + 1 PC/Fq (q ) √ −g 1 q q hC/Fq φ2(C/Fq) = √ −1/2 , (5.5) 2 q − 1 PC/Fq (−q ) −2 X ZC/Fq (γ ) γ φ3(C/ q) = . F Z 0 (γ−1) γ − 1 γ C/Fq For Question 5.6, we study in Chapter 7 the average proportion of hyperelliptic curves C ∈ H2g+1,qn satisfying P´olya’s conjecture as the finite field Fq grows larger. 62 P´olya’s Conjecture in Function Fields

Much as we do with the Mertens conjecture, we are able to relate this average proportion to the Haar measure of the pullback of the region where a certain function of random matrices is nonnegative, which we are then able to calculate explicitly for low values of g. We find that most curves in this family do not satisfy P´olya’s conjecture.

Theorem 5.9 (cf. Theorem 1.8). Fix 1 ≤ g ≤ 2, and suppose that the char- acteristic of Fq is odd. Then as n tends to infinity, most hyperelliptic curves C ∈ H2g+1,qn do not satisfy P´olya’sconjecture for C/Fqn . Chapter 6

Local P´olya Conjectures

6.1 An Explicit Expression for LC/Fq (X) We obtain an explicit description for the summatory function of the Liouville function in function fields by studying the Dirichlet series

X λC/ q (D) F , N Ds D≥0 which converges absolutely for <(s) > 1. As λC/Fq (D) is completely multiplicative and satisfies λ(P ) = −1 for a prime divisor P of C, this has the Euler product expansion

X λC/ q (D) Y 1 F = N Ds 1 + N P −s D≥0 P for <(s) > 1, which upon comparing Euler products leads us to the identity

X λC/ q (D) ζC/ q (2s) F = F , (6.1) N Ds ζ (s) D≥0 C/Fq which is valid for all <(s) > 1. On the other hand, note that for <(s) > 1, we may rearrange this Dirichlet series instead to be of the form

∞ X λC/ q (D) X λC/ q (D) X 1 X F = F = λ (D). (6.2) N Ds qdeg(D)s qNs C/Fq D≥0 D≥0 N=0 deg(D)=N

We will determine an expression for the coefficients of the Dirichlet series for

ζC/Fq (2s)/ζC/Fq (s) using (2.1) and compare coefficients in order to find a formula PX−1 P for LC/Fq (X) = N=0 deg(D)=N λC/Fq (D). Along the way, we will require the following lemma.

63 64 Local P´olya Conjectures

Lemma 6.1 ([24, Proposition 8.16]). Let Fq2 be the quadratic field extension of Fq, which is unique up to isomorphism, and let P (u) C/Fq2 ZC/ (u) = Fq2 (1 − u)(1 − q2u) be the zeta function of C/Fq2 . Then for all u ∈ C,

P u2
= P (u)P (−u). C/Fq2 C/Fq C/Fq

Consequently, h C/Fq2 PC/Fq (−1) = , hC/Fq where hC/Fq = PC/Fq (1) is the class number of the function field C/Fq.

Proposition 6.2 (cf. Proposition 2.4). Let C be a nonsingular projective curve −1 over Fq of genus g ≥ 0, and suppose that all of the zeroes γ of ZC/Fq (u) are simple. Then for each N ≥ 0,

X λC/Fq (D) deg(D)=N √ −g √ −g 1 q − 1 q hC/Fq (N+1)/2 N+1 1 q + 1 q hC/Fq (N+1)/2 = − √ −1/2 q − (−1) √ −1/2 q 2 q + 1 PC/Fq (q ) 2 q − 1 PC/Fq (−q ) −2 2 X ZC/Fq (γ ) N+1 N+1 q + 1 hC/Fq − 0 −1 γ + (−1) + R(N, q, g, T ), (6.3) ZC/ (γ ) q − 1 hC/ γ Fq Fq2 where the sum is over the inverse zeroes of ZC/Fq (u), T > 0 is sufficiently small, and the error term R(N, q, g, T ) satisfies R(N, q, g, T ) = 0 if N ≥ max{2g−1, 0}.

T Proof. We let CT = {z ∈ C : |z| = q } for T > 0, and we study the contour integral I 2 1 1 ZC/Fq (u ) N+1 du. 2πi CT u ZC/Fq (u) 2 There are two different identities for ZC/Fq (u ) /ZC/Fq (u), obtainable from (6.1) and (2.1) and from (6.2), which give the identities

2 2g 2 ZC/ q (u ) 1 − qu Y 1 − γju F = √ √ , (6.4) Z (u) 1 − qu
1 + qu
(1 + u) 1 − γ u C/Fq j=1 j 2 ∞ ZC/Fq (u ) X N X = u λC/Fq (D), (6.5) ZC/ (u) Fq N=0 deg(D)=N 6.1 An Explicit Expression for LC/Fq (X) 65

−1/2 −1 −1 with the first identity valid for all u ∈ C \ {±q , −1, γ1 , . . . , γ2g }, and the −1 −1/2
second identity valid for all |u| < q . If ZC/Fq ±q 6= 0, the singularities −1/2 −1 of the integrand inside CT occur at u = 0, u = −1, u = ±q , and u = γ −1 for each zero γ of ZC/Fq (u); note that the assumption of the simplicity of the −1/2 zeroes of ZC/Fq (u) means that none of these zeroes can occur at u = ±q . At the singularity u = 0, we have by (6.5) that

2 1 ZC/Fq (u ) X Res N+1 = λC/Fq (D). u=0 u ZC/ (u) Fq deg(D)=N

At the singularity u = −1, (6.4) and Lemma 6.1 show that

1 Z (u2) q + 1 P (1) q + 1 h 2 Res C/Fq = (−1)N C/Fq = (−1)N C/Fq . u=−1 uN+1 Z (u) q − 1 P (−1) q − 1 h C/Fq C/Fq C/Fq2

At the singularities u = ±q−1/2,

1 Z (u2) Res C/Fq −1/2 N+1 u=±q u ZC/Fq (u)   2 1 1 1 − qu PC/ (u ) = lim u ∓ √ √ √ Fq −1/2 N+1

u→±q q u 1 − qu 1 + qu (1 + u) PC/Fq (u) √ −g N+1 1 q ∓ 1 q hC/Fq (N+1)/2 = (±1) √ −1/2 q , 2 q ± 1 PC/Fq (±q ) where we have used the fact that 1 P = q−gP (1) = q−gh , C/Fq q C/Fq C/Fq which follows from the functional equation (2.2) for ZC/Fq (u). Finally, as ZC/Fq (u) has a simple zero at each γ−1,

2 −2 1 ZC/Fq (u ) ZC/Fq (γ ) N+1 Res = 0 γ . u=γ−1 N+1 −1 u ZC/Fq (u) ZC/Fq (γ ) So by Cauchy’s residue theorem,

I 2 1 1 ZC/Fq (u ) N+1 du 2πi CT u ZC/Fq (u) √ −g √ −g 1 q − 1 q hC/Fq (N+1)/2 N+1 1 q + 1 q hC/Fq (N+1)/2 = √ −1/2 q + (−1) √ −1/2 q 2 q + 1 PC/Fq (q ) 2 q − 1 PC/Fq (−q ) −2 2 X ZC/Fq (γ ) N+1 N+1 q + 1 hC/Fq X + 0 −1 γ − (−1) + λC/Fq (D), (6.6) ZC/ (γ ) q − 1 hC/ γ Fq Fq2 deg(D)=N 66 Local P´olya Conjectures

which yields (6.3), with

I 2 1 1 ZC/Fq (u ) R(N, q, g, T ) = N+1 du. 2πi CT u ZC/Fq (u)

T √ By (6.4) and the fact that |u| = q and |γj| = q,

I 2 1 1 ZC/Fq (u ) |R(N, q, g, T )| ≤ N+1 |du| 2π CT u ZC/Fq (u) q1+T + 1
q1/2+2T + 1
2g ≤ q−NT . (6.7) (qT − 1) (q1+2T − 1) (q1/2+T − 1)2g

If g ≥ 1 and N = 2(g − 1), then the right-hand side tends to one as T tends to infinity, while for all g ≥ 0 and for all N ≥ max{2g − 1, 0}, the right-hand side tends to zero as T tends to infinity. As the right-hand side of (6.6) is independent of T , we may take the limit as T tends to infinity on both sides, which implies that the left-hand side of (6.6) has absolute value at most than one if g ≥ 1 and N = 2(g − 1), while for N ≥ max{2g − 1, 0} the left-hand side of (6.6) is equal to zero.

We obtain an explicit expression for LC/Fq (X) by summing (6.3) over all 0 ≤ N ≤ X − 1 and evaluating the resulting geometric progressions, which yields

√ −g √ −g 1 q q hC/Fq X/2 X 1 q q hC/Fq X/2 LC/Fq (X) = − √ −1/2 q − (−1) √ −1/2 q 2 q + 1 PC/Fq (q ) 2 q − 1 PC/Fq (−q ) −2 X ZC/ q (γ ) γ − F γX + R (q, g, T ), (6.8) Z 0 (γ−1) γ − 1 X γ C/Fq where

−g −2 q q hC/ q X ZC/ q (γ ) γ R (q, g, T ) = F + F X q − 1 P (q−1/2) Z 0 (γ−1) γ − 1 C/Fq γ C/Fq X 2 X−1 (−1) − 1 q + 1 hC/ q X + F + R(N, q, g, T ). (6.9) 2 q − 1 hC/ Fq2 N=0

In particular, after fixing T > 0 we have that RX (q, g, T ) = Oq,g(1) as X tends to infinity. The expression (6.8) suggests that LC/Fq (X) grows at a rate comparable to qX/2. Indeed, by using the fact that each simple inverse zero can be written √ in the form γ = qeiθ(γ) with −π < θ(γ) < π, we can convert (6.8) into an X/2 asymptotic equation for LC/Fq (X)/q . 6.1 An Explicit Expression for LC/Fq (X) 67

Corollary 6.3. Let C be a nonsingular projective curve over Fq of genus g ≥ 0, −1 and suppose that all of the zeroes γ of ZC/Fq (u) are simple. Then as X tends to infinity,

√ −g √ −g LC/Fq (X) 1 q q hC/Fq X 1 q q hC/Fq X/2 = − √ −1/2 − (−1) √ −1/2 q 2 q + 1 PC/Fq (q ) 2 q − 1 PC/Fq (−q ) −2   X ZC/ q (γ ) γ 1 − F eiXθ(γ) + O . (6.10) Z 0 (γ−1) γ − 1 q,g qX/2 γ C/Fq −s Recalling that ZC/Fq (u) = ζC/Fq (s) with u = q , we see that we can rewrite (6.10) as

LC/Fq (X) qX/2 √
−g √
−g 1 q q + 1 q hC/Fq X 1 q q − 1 q hC/Fq = 2 − (−1) 2 2 (q − 1) ζC/Fq (1/2) 2 (q − 1) ζC/Fq (1/2 + πi/ log q) iX=(ρ)   X ζC/ q (2ρ) q 1 + log q F + O , ζ 0(ρ) qρ − 1 q,g qX/2 ρ C/Fq where the sum is over the 2g zeroes ρ = 1/2 + i=(ρ) of ζC/Fq (s) that lie in the range −π/ log q < =(ρ) < π/ log q. On the other hand, we have the explicit √ expression (5.2) for L(x)/ x: L(x) 1 X ζ(2ρ) xi=(ρ)  1  √ = + + O √ . x ζ(1/2) ζ0(ρ) ρ x ρ √ We see many similarities between our explicit expressions (5.2) for L(x)/ x and X/2 (6.10) for LC/Fq (X)/q , though several new features appear in the function field case, most notably additional leading terms. In the number field case, this is merely the reciprocal of the zeta function evaluated at the critical point s = 1/2. In the function field case, the leading term is no longer constant: here we have an additional critical point at s = 1/2 + πi/ log q, halfway up the critical line (recalling that ζC/Fq (s) is periodic with period 2πi/ log q), which leads to X/2 oscillations of LC/Fq (X)/q according to whether X is even or odd. Finally, the coefficients of these leading terms are heavily dependent on the genus of C and on the size q of the finite field Fq over which C is defined. We are interested in using the explicit expression (6.10) to study sign changes, or lack thereof, of LC/Fq (X). To understand the behaviour of LC/Fq (X) when the genus of the curve C is greater than zero, we must first determine the signs of three important quantities that appear in (6.3): the class number hC/Fq and the −1/2
values PC/Fq ±q . 68 Local P´olya Conjectures

Lemma 6.4 ([24, Proposition 5.11]). For the class number hC/Fq of the function field C/Fq, we have the bounds

√ 2g √ 2g ( q − 1) ≤ hC/Fq ≤ ( q + 1) .

−g In particular, q hC/Fq is always positive, and we have the asymptotic  1  q−gh = 1 + O √ (6.11) C/Fq g q as q tends to infinity.

−1/2
Next, we observe we can write PC/Fq ±q explicitly in terms of the inverse √ iθ(γ) zeroes γ = qe of ZC/Fq (u). Using the fact that the 2g inverse zeroes γ satisfy −1 γj+g = qγj for 1 ≤ j ≤ g, we see that   g    √  g 1 Y γj q Y P ±√ = 1 ∓ √ 1 ∓ = 1 ∓ eiθ(γj )
1 ∓ e−iθ(γj )
. C/Fq q q γ j=1 j j=1 From this, standard trigonometric identities allow us to show the following.

Lemma 6.5. We have that g  1  Y P √ = 2g (1 − cos θ(γ )), C/Fq q j j=1 g  1  Y P −√ = 2g (1 + cos θ(γ )). C/Fq q j j=1

−1/2
In particular, PC/Fq ±q are both always nonnegative, and are strictly positive −1/2
√ when ZC/Fq ±q 6= 0: that is, when ± q are not inverse zeroes of ZC/Fq (u). So by using the triangle inequality on (6.10), we obtain the following bounds X/2 for LC/Fq (X)/q ; in particular, we prove part of Theorem 5.7 in showing that if ZC/Fq (u) has simple zeroes, then P´olya’s conjecture for C/Fq is true when the inequality (5.4) holds.

Corollary 6.6. Let C be a nonsingular projective curve over Fq of genus g ≥ 1, −1 and suppose that all of the zeroes γ of ZC/Fq (u) are simple. Let

+ LC/Fq (X) B (C/Fq) = lim sup X/2 , X→∞ q

− LC/Fq (X) B (C/Fq) = lim inf . X→∞ qX/2 6.1 An Explicit Expression for LC/Fq (X) 69

Then we have the bounds

√ −g √ −g + 1 q q hC/Fq 1 q q hC/Fq B (C/Fq) ≤ − √ −1/2 + √ −1/2 2 q + 1 PC/Fq (q ) 2 q − 1 PC/Fq (−q ) −2 X ZC/Fq (γ ) γ + , Z 0 (γ−1) γ − 1 γ C/Fq

√ −g √ −g − 1 q q hC/Fq 1 q q hC/Fq B (C/Fq) ≥ − √ −1/2 − √ −1/2 2 q + 1 PC/Fq (q ) 2 q − 1 PC/Fq (−q ) −2 X ZC/Fq (γ ) γ − . Z 0 (γ−1) γ − 1 γ C/Fq

−1 The assumption that all of the zeroes γ of ZC/Fq (u) are simple implies that X/2 the bounds above are finite, that is, that LC/Fq (X)/q is bounded. This high- X/2 √ lights a notable difference in the behaviour of LC/Fq (X)/q to that of L(x)/ x. While the former is bounded as X tends to infinity, the latter is conjectured to grow unboundedly in both the positive and negative directions as x tends to in- √ finity. The key difference here is that the explicit expression (5.2) for L(x)/ x X/2 involves an infinite sum over the zeroes of ζ(s), while for LC/Fq (X)/q the anal- ogous sum has only finitely many terms, as there only finitely many zeroes of

ZC/Fq (u). Next, we show that the bounds in Corollary 6.6 are strict when C satisfies

LI, from which it follows that when C satisfies LI, P´olya’s conjecture for C/Fq is true if and only if the inequality (5.4) holds.

Theorem 6.7 (cf. Theorem 2.6). Suppose that C satisfies LI. Then

√ −g √ −g + 1 q q hC/Fq 1 q q hC/Fq B (C/Fq) = − √ −1/2 + √ −1/2 2 q + 1 PC/Fq (q ) 2 q − 1 PC/Fq (−q ) −2 X ZC/Fq (γ ) γ + , Z 0 (γ−1) γ − 1 γ C/Fq

√ −g √ −g − 1 q q hC/Fq 1 q q hC/Fq B (C/Fq) = − √ −1/2 − √ −1/2 2 q + 1 PC/Fq (q ) 2 q − 1 PC/Fq (−q ) −2 X ZC/Fq (γ ) γ − . Z 0 (γ−1) γ − 1 γ C/Fq 70 Local P´olya Conjectures

Proof. We have that

−2 g −2
! X ZC/ q (γ ) γ X ZC/ q γj γj F eiXθ(γ) = 2< F eiXθ(γj ) , Z 0 (γ−1) γ − 1 Z 0 γ−1
γ − 1 γ C/Fq j=1 C/Fq j j so that

√ −g √ −g LC/Fq (X) 1 q q hC/Fq X 1 q q hC/Fq X/2 = − √ −1/2 − (−1) √ −1/2 q 2 q + 1 PC/Fq (q ) 2 q − 1 PC/Fq (−q ) g −2
!   X ZC/ q γj γj 1 − 2< F eiXθ(γj ) + O . Z 0 γ−1
γ − 1 q,g qX/2 j=1 C/Fq j j The assumption that C satisfies LI along with the Kronecker–Weyl theorem in- form us that the set

 πiX iXθ(γ1) iXθ(γg)
g+1 e , e , . . . , e ∈ T : X ∈ N

g is equidistributed in {±1}×T . This implies the existence of a subsequence (Xm) of N such that √ −g √ −g LC/ (Xm) 1 q q hC/ 1 q q hC/ lim Fq = − √ Fq + √ Fq m→∞ Xm/2 −1/2 −1/2 q 2 q + 1 PC/Fq (q ) 2 q − 1 PC/Fq (−q ) −2 X ZC/Fq (γ ) γ + , Z 0 (γ−1) γ − 1 γ C/Fq

+ and hence the upper bound for B (C/Fq) is sharp. Similarly, a subsequence − exists that ensures that the lower bound for B (C/Fq) is also sharp.

We remark that we may also study the behaviour of LC/Fq (X) when ZC/Fq (u) has multiple zeroes, but that the situation is not so easily resolved as with

MC/Fq (X), as analysed in Section 2.2. The key difference is the behaviour of −1/2 LC/Fq (X) when ZC/Fq (u) has a zero of multiple order at u = q . If there is a zero elsewhere of higher order than the zero at q−1/2, however, then one can mimic the proofs of Propositions 2.12 and 2.13 to show that LC/Fq (X) changes sign infinitely often; we omit the details.

6.2 Examples in Low Genus

1 When g = 0, so that C = P and hence that C/Fq = Fq(t), it is particularly easy to determine the limiting behaviour of LC/Fq (X) via (6.8) and (6.9), as −1/2
hC/Fq ,PC/Fq ±q are all equal to 1 and there are no inverse zeroes γ. 6.2 Examples in Low Genus 71

Proposition 6.8 (cf. Proposition 2.3). Let g = 0. Then q(X+1)/2 − 1  if X is odd,  q − 1 LC/Fq (X) = (6.12)  qX/2+1 − q − if X is even.  q − 1

Consequently, LC/Fq (X) changes sign infinitely often, and √ q B+(C/ ) = , Fq q − 1 q B−(C/ ) = − . Fq q − 1

In particular, P´olya’sconjecture for C/Fq is false. Alternatively, one can prove the identity (6.12) by noting that from (6.4) and (6.5), we have that for |u| < q−1, ∞ X X (1 − qu) uN λ (D) = C/Fq (1 + u)(1 − qu2) N=0 deg(D)=N ∞ ∞ X X = (1 − qu) (−1)AuA qBu2B A=0 B=0   ∞ X N  X A B/2 X A B/2+1 = u  (−1) q − (−1) q . N=0 A+B=N A+B=N−1 B even B even N P Equating coefficients of u , we find that deg(D)=0 λC/Fq (D) = 1, while for N ≥ 1,  N/2+1 2q − q − 1  if N is even, X  q − 1 λC/Fq (D) =  q(N+3)/2 + q(N+1)/2 − q − 1 deg(D)=N − if N is odd.  q − 1 Summing over all 0 ≤ N ≤ X − 1 yields (6.12). Similar results can be determined when g = 1, so that C is an elliptic curve over a finite field Fq. When C satisfies LI, we have the following result.

Proposition 6.9. Let C be an elliptic curve over Fq, and suppose that C satisfies LI. Then √ r q q + 1 − a 2 q + 1 − ap B+(C/ ) = − (a − 2) + q2 + q + 3aq − a3, Fq q − 1 4q − a2 4q − a2 q + 1 + a √ r q q + 1 − a 2 q + 1 − ap B−(C/ ) = − (2q − a) − q2 + q + 3aq − a3, Fq q − 1 4q − a2 4q − a2 q + 1 + a where a is the trace of the Frobenius endomorphism. 72 Local P´olya Conjectures

2 Proof. Using the fact that PC/Fq (u) = 1 − au + qu , so that hC/Fq = q + 1 − a and −1/2
−1/2 PC/Fq ±q = 2 ∓ aq , we have that

√ −g √ −g √ 1 q q hC/Fq 1 q q hC/Fq q q + 1 − a − √ −1/2 + √ −1/2 = − 2 (a − 2), 2 q + 1 PC/Fq (q ) 2 q − 1 PC/Fq (−q ) q − 1 4q − a √ −g √ −g √ 1 q q hC/Fq 1 q q hC/Fq q q + 1 − a − √ −1/2 − √ −1/2 = − 2 (2q − a). 2 q + 1 PC/Fq (q ) 2 q − 1 PC/Fq (−q ) q − 1 4q − a √ Next, the fact that γ = qeiθ(γ) implies that

(1 − γ−1) (1 − γγ−2) Z γ−2
= C/Fq (1 − γ−2) (1 − qγ−2) γ2 − γ = (γ − γ)(γ + 1) √ √ q cos 2θ(γ) − q cos θ(γ) + iq sin 2θ(γ) + i q sin θ(γ) = √ √ √ . 2i q sin θ(γ) q cos θ(γ) + 1 + i q sin θ(γ)

So by the Pythagorean trigonometric identity and the cosine angle-difference and triple-angle formulæ, s 2 3/2 3/2 3 −2
1 q + q + 6q cos θ(γ) − 8q cos θ(γ) ZC/ q γ = √ √ F 2 q sin θ(γ) q + 1 + 2 q cos θ(γ) s q2 + q + 3aq − a3 = , (4q − a2)(q + 1 + a) as √ 2 q cos θ(γ) = a, √ p 2 q sin θ(γ) = 4q − a2.

Finally, the proof of Proposition 3.3 shows that r 1 γ q + 1 − a 2 0 −1 = 2 2 , ZC/Fq (γ ) γ − 1 4q − a and the result now follows from Theorem 6.7.

Via Mathematica, we have found that for q a prime power and a an integer √ satisfying |a| < 2 q, the function √ r q q + 1 − a 2 q + 1 − ap − (a − 2) + q2 + q + 3aq − a3 q − 1 4q − a2 4q − a2 q + 1 + a 6.2 Examples in Low Genus 73

is always positive; consequently, P´olya’s conjecture is always false for C/Fq when the elliptic curve C satisfies LI. In spite of this, when q is a perfect square, we are ensured an elliptic curve

C over Fq for which P´olya’s conjecture holds, via the curve whose trace of the √ −1/2 Frobenius a is equal to 2 q, so that ZC/Fq (u) has a zero of order two at u = q .

Proposition 6.10. Let C be an elliptic curve over a finite field Fq of character- −1/2 istic p, and suppose that ZC/ (u) has a zero of multiple order at u = q , so √ Fq that q = pm with a = ±2 q, where m is even. Then

√
2 LC/ (X) 1 q − 1 Fq = − √ √ X2 + O (X). qX/2 4 q q + 1 q

In particular, P´olya’sconjecture holds for C/Fq. √ √ Proof. When a = 2 q, we have that γ = γ = q, and so the proof of Proposition 6.2 shows that for N ≥ 0,

2 X 1 ZC/Fq (u ) N q − 1 (N+1)/2 λC/Fq (D) = − Res N+1 + (−1) q u=q−1/2 u ZC/ (u) 8q deg(D)=N Fq √ 2 I 2 N q + 1 q − 1 1 1 ZC/Fq (u ) − (−1) √ + N+1 du, q − 1 q + 1 2πi CT u ZC/Fq (u) with the last term equal to zero for N ≥ 1. Now by the binomial theorem,

∞ k 1 X N + k  1  = q(N+1)/2 (−1)k qk/2 u − √ , uN+1 k q k=0 whereas √ 2 Z (u2) (1 − qu) 1 − qu2
C/Fq = √
√
3 ZC/Fq (u) (1 + u) 1 + qu 1 − qu √ 3 ! 1 q − 1
 1 −3  1 −2 = √ u − √ + O u − √ 2q2 q + 1 q q q in a neighbourhood of u = q−1/2, and as for fixed k, N + k N k = + O N k−1
k k! in the large N limit, we find via Laurent series about u = q−1/2 that

2 √
3 1 ZC/Fq (u ) 1 q − 1 2 (N+1)/2 (N+1)/2
Res = √ N q + Oq Nq −1/2 N+1 u=q u ZC/Fq (u) 4q q + 1 74 Local P´olya Conjectures

as N grows large. Thus

√
3 X 1 q − 1 λ (D) = − √ N 2q(N+1)/2 + O Nq(N+1)/2
, C/Fq 4q q + 1 q deg(D)=N and summing over all 0 ≤ N ≤ X − 1 and then dividing through by qX/2 yields the result.

X/2 6.3 The Limiting Distribution of LC/Fq (X)/q This section mimics Section 2.3 in determining the natural density of the set of natural numbers for which LC/Fq (X) ≤ 0. From Corollary 6.3, for any nonsingular projective curve C over Fq of genus g ≥ 1 we may write L (X) C/Fq = E (X) + ε (X), qX/2 C/Fq;λ C/Fq;λ where

√ −g √ −g 1 q q hC/Fq X 1 q q hC/Fq EC/Fq;λ(X) = − √ −1/2 − (−1) √ −1/2 2 q + 1 PC/Fq (q ) 2 q − 1 PC/Fq (−q ) −2 X ZC/ q (γ ) γ − F eiXθ(γ), Z 0 (γ−1) γ − 1 γ C/Fq  1  ε (X) = O , C/Fq;λ q,g qX/2

−1 provided that all of the zeroes γ of ZC/Fq (u) are simple. Using this, we may X/2 prove the existence of a limiting distribution of LC/Fq (X)/q by first construct- ing the limiting distribution of EC/Fq;λ(X).

Lemma 6.11 (cf. Lemma 2.15). Let C be a nonsingular projective curve over −1 Fq of genus g ≥ 1, and suppose that all of the zeroes γ of ZC/Fq (u) are simple.

There exists a probability measure νC/Fq;λ on R that satisfies

Y Z 1 X
lim f EC/ ;λ(X) = f(x) dνC/ ;λ(x) Y →∞ Y Fq Fq X=1 R for all continuous functions f on R.

Proof. The proof is essentially the same as that of Lemma 2.15, though notably the subtorus H is different. This time, we apply the Kronecker–Weyl theorem X/2 6.3 The Limiting Distribution of LC/Fq (X)/q 75

with t0 = π/2, tj = θ(γj)/2π for 1 ≤ j ≤ g, in order to deduce the existence of a subtorus H ⊂ Tg+1 satisfying

1 Y Z X πiX iXθ(γ1) iXθ(γg)
lim h e , e , . . . , e = h(z) dµH (z) Y →∞ Y X=1 H for every continuous function h on Tg+1. We then define the probability measure

νC/Fq;λ on R by

νC/Fq;λ(B) = µH (Be) for each Borel set B ⊂ R, where

( g ! ) X j Be = (z0, z1, . . . , zg) ∈ H : −φ1 − φ2z0 − 2< φ3zj ∈ B . j=1

Here for brevity’s sake φ1 = φ1(C/Fq) and φ2 = φ2(C/Fq) are as in (5.5), while for 1 ≤ j ≤ g, −2 iθ(γj ) j j ZC/Fq (γj ) γje φ3 = φ3(C/Fq) = 0 −1 , (6.13) ZC/Fq (γj ) γj − 1 so that g X j φ3 = φ3(C/Fq) = 2 φ3 . j=1 g+1 For f a bounded continuous function on R, h(z0, z1, . . . , zg) on T is defined by

g !! X j h(z0, z1, . . . , zg) = f −φ1 − φ2z0 − 2< φ3zj , j=1 so that h is continuous on Tg+1, and consequently Z Z f(x) dνC/Fq;λ(x) = h(z0, z1, . . . , zg) dµH (z0, z1, . . . , zg) R H Y 1 X = lim h eπiX , eiXθ(γ1), . . . , eiXθ(γg)
Y →∞ Y X=1 Y 1 X
= lim f EC/ ;λ(X) . Y →∞ Y Fq X=1 The proof of the next result is essentially unchanged from the proof of Propo- sition 2.18.

Proposition 6.12 (cf. Proposition 2.18). Let C be a nonsingular projective curve −1 over Fq of genus g ≥ 1, and suppose that all of the zeroes γ of ZC/Fq (u) are 76 Local P´olya Conjectures

X/2 simple. The function LC/Fq (X)/q has a limiting distribution νC/Fq;λ on R.

That is, there exists a probability measure νC/Fq;λ on R such that

Y   Z 1 X LC/Fq (X) lim f = f(x) dνC/ ;λ(x) Y →∞ Y qX/2 Fq X=1 R for all bounded continuous functions f on R.

We are now able to prove Theorem 5.8.

Proof of Theorem 5.8. The Portmanteau Theorem in conjunction with Proposi- tion 6.12 implies that   1 LC/Fq (X) lim # 1 ≤ X ≤ Y : ∈ B = νC/ ;λ(B) Y →∞ Y qX/2 Fq

for every Borel set B ⊂ R whose boundary has νC/Fq;λ-measure zero, and hence d(PC/Fq;λ) exists and is equal to νC/Fq;λ((−∞, 0]) if νC/Fq;λ({0}) = 0. This follows as the assumption that C satisfies LI implies that H = {±1} × Tg, that is, that H is the union of two disjoint subtori, and hence the normalised Haar measure on H is half the Lebesgue measure on each subtorus. Thus for a Borel set B ⊂ R,

g ! ! 1 X ν (B) = m −φ − φ − 2< φj e2πiθj ∈ B C/Fq;λ 2 1 2 3 j=1 g ! ! 1 X + m −φ + φ − 2< φj e2πiθj ∈ B 2 1 2 3 j=1 g ! 1 X j = m −φ1 − φ2 − 2 φ cos(2πθj) ∈ B 2 3 j=1 g ! 1 X j + m −φ1 + φ2 − 2 φ cos(2πθj) ∈ B . 2 3 j=1

Pg j g The function j=1 φ3 cos(2πθj) is real analytic on [0, 1] and not uniformly Pg j  constant, so m j=1 φ3 cos(2πθj) = c = 0 for all c ∈ R, from which it follows that νC/Fq;λ is atomless. Finally, the fact that

g ! g ! X j X j m φ3 cos(2πθj) ≥ c = m φ3 cos(2πθj) ≤ −c j=1 j=1 X/2 6.3 The Limiting Distribution of LC/Fq (X)/q 77

for any c ∈ R implies that

g ! 1 1 X j d(PC/ q;λ) = + m −φ1 − φ2 ≤ 2 φ cos(2πθj) ≤ φ1 − φ2 F 2 2 3 j=1 g ! 1 X j = 1 − m 2 φ cos(2πθj) < −φ1 − φ2 2 3 j=1 g ! 1 X j − m 2 φ cos(2πθj) < −φ1 + φ2 . 2 3 j=1

From this, it is clear that  1/2 if −φ1(C/Fq) + φ2(C/Fq) ≥ φ3(C/Fq), d(PC/Fq;λ) = 1 if −φ1(C/Fq) + φ2(C/Fq) ≤ −φ3(C/Fq).

If −φ3 < −φ1 + φ2 < φ3, then there exists an open neighbourhood of (0,..., 0) ∈ g [0, 1] such that for all (θ1, . . . , θg) in this neighbourhood,

g X j |−φ1 + φ2| ≤ 2 φ3 cos(2πθj) ≤ φ1 + φ2 j=1 and consequently 1/2 < d(PC/Fq;λ) < 1. 78 Local P´olya Conjectures Chapter 7

Global P´olya Conjectures

7.1 Averages over Families of Curves

We wish to find the average, as the finite field Fq grows larger, of the number of curves for which P´olya’s conjecture is true. Our first step is determine expressions for φk(C/Fq), 1 ≤ k ≤ 3, in terms of Zϑ(C/Fqn )(θ), the characteristic polynomial # of ϑ(C/Fqn ) ∈ USp2g(C) , in the large q limit. The resulting expressions involve particular functions related to ZU (θ); these are the functions 1 1 ψ1(U) = , 2 ZU (0) 1 1 ψ2(U) = , 2 ZU (π) 2g 1 X | cosec θj| ψ (U) = . 3 2 |Z 0(θ )| j=1 U j

We also define the functions

ψ±(U) = −ψ1(U) ± ψ2(U) ± ψ3(U).

Note that ψ1, ψ2, ψ3 are always nonnegative, but that they can be infinite: ψ1(U) blows up if U has an eigenvalue equal to 1 (which is necessarily a repeated eigen- value), while ψ2(U) blows up when U has an eigenvalue equal to −1 (which must also be a repeated eigenvalue), and ψ3(U) blows up whenever U has a repeated eigenvalue. Recall, however, that the set of matrices in USp2g(C) with repeated eigenvalues has measure zero with respect to the normalised Haar measure on

USp2g(C).

79 80 Global P´olya Conjectures

−1 Lemma 7.1 (cf. Lemma 4.2). Suppose that all of the zeroes γ of ZC/Fq (u) are simple. Then for 1 ≤ k ≤ 3,  1  φ (C/ ) = ψ (ϑ(C/ )) + O √ ψ (ϑ(C/ )) . k Fq k Fq g q k Fq

Consequently,  1  B+(C/ ) = ψ (ϑ(C/ )) + O −√ ψ (ϑ(C/ )) . Fq + Fq g q − Fq

Proof. From the definitions of φ1(C/Fq) and φ2(C/Fq) and from (4.2), we have that

√ −g 1 q q hC/Fq φ1(C/Fq) = √ , 2 q + 1 Zϑ(C/Fq)(0) √ −g 1 q q hC/Fq φ2(C/Fq) = √ . 2 q − 1 Zϑ(C/Fq)(π)

The desired identities for φ1(C/Fq) and φ2(C/Fq) then follow from the asymptotic (6.11). For φ3(C/Fq), we have by (2.1), (4.2), and (4.3) that

−2 −2 3 ZC/Fq (γ ) γ PC/Fq (γ ) γ − 1 γ 0 −1 = 0 −1 2 ZC/Fq (γ ) γ − 1 PC/Fq (γ ) γ − 1 γ − γ −2 2 PC/Fq (γ ) γ − 1 iγ = 0 2 . Zϑ(C/Fq) (θ(γ)) γ − 1 γ − γ √ iθ(γj ) As γj = qe for each 1 ≤ j ≤ 2g, we may rewrite this as

−2 ZC/Fq (γ ) γ 0 −1 ZC/Fq (γ ) γ − 1 Q2g −1/2 i(θ(γj )−2θ(γ))
√ −iθ(γ) 2iθ(γ) j=1 1 − q e qe − 1 iqe = 0 2iθ(γ) √ iθ(γ) √ −iθ(γ) Zϑ(C/Fq) (θ(γ)) qe − 1 qe − qe 1 e−iθ(γ) cosec θ(γ) c e−iθ(γ) cosec θ(γ) = 0 + √ 0 2 Zϑ(C/Fq) (θ(γ)) q Zϑ(C/Fq) (θ(γ)) for some coefficient c ∈ C dependent on q, g, θ(γ), θ(γ1), . . . , θ(γg) and uniformly bounded in q, θ(γ), θ(γ1), . . . , θ(γg). This yields the asymptotic

−2   ZC/Fq (γ ) γ 1 | cosec θ(γ)| 1 | cosec θ(γ)| 0 −1 = 0 + Og √ 0 , ZC/Fq (γ ) γ − 1 2 |Zϑ(C/Fq) (θ(γ))| q |Zϑ(C/Fq) (θ(γ))| and by summing over all inverse zeroes γ, we obtain the desired identity for

φ3(C/Fq). 7.1 Averages over Families of Curves 81

Lemma 7.2 (cf. Lemma 4.8). Let B be an interval in R. Then the boundaries of the sets  U ∈ USp2g(C): ψ±(U) ∈ B have Haar measure zero.

Proof. By (4.1), we have that

g g 1 Y 1 1 Y 1 ψ (θ , . . . , θ ) = − ± ± 1 g 2g+1 (1 − cos θ ) 2g+1 (1 + cos θ ) j=1 j j=1 j g g 1 X 2 Y 1 ± g cosec θj . 2 |cos θk − cos θj| j=1 k=1 k6=j

It suffices to show that for each permutation σ of {1, . . . , g} and for each c ∈ R, the sets

 g (θσ(1), . . . , θσ(g)) ∈ [0, π] : ψ±(θσ(1), . . . , θσ(g)) = c, 0 < θ< . . . < θg < π have Lebesgue measure zero, and this is true as when 0 < θ1 < . . . < θg < π, the functions ψ±(θσ(1), . . . , θσ(g)) are real analytic and non-uniformly constant.

Lemma 7.3 (cf. Lemma 4.9). For all g ≥ 1, the function ψ− on USp2g(C) is integrable and satisfies the bounds Z 2(g−1) −1 − 2 ≤ ψ−(U) dµHaar(U) ≤ −1. USp2g(C) The proof of this result follows from the following two lemmata, together with 2(g−1) the bound 0 ≤ |ZU (θ)| ≤ 2 for all U ∈ USp2(g−1)(C) and θ ∈ [0, π]. Lemma 7.4 (Keating–Snaith [14]). We have the identities Z Z 1 ψ (U) dµ (U) = ψ (U) dµ (U) = . 1 Haar 2 Haar 2 USp2g(C) USp2g(C) Proof. Keating and Snaith show that [14, §2.1 Equation (10)] Z 1 dµ (U) = 1. Z (0) Haar USp2g(C) U

The Haar measure is invariant under left multiplication by matrices V ∈ USp2g(C), so by taking V = −I, so that ZVU (0) = det(I + U) = ZU (π), we find that Z 1 Z 1 dµ (U) = dµ (U) = 1. Z (π) Haar Z (0) Haar USp2g(C) U USp2g(C) U 82 Global P´olya Conjectures

Lemma 7.5 (cf. Lemma 4.10). For g = 1, we have that Z ψ3(U) dµHaar(U) = 1, USp2(C) while for g ≥ 2, we have the identity Z 1 Z π Z ψ (U) dµ (U) = |Z (θ)| dµ (U) dθ. 3 Haar π U Haar USp2g(C) 0 USp2(g−1)(C) Proof. The g = 1 case is trivial. For g ≥ 2, we differentiate (4.1) in order to find that 2g g g 1 X | cosec θj| 1 X 2 Y 1 0 = g cosec θj . 2 |ZU (θj)| 2 |cos θk − cos θj| j=1 j=1 k=1 k6=j By the Weyl integration formula,

Z 2g 1 X | cosec θj| 0 dµHaar(U) 2 |ZU (θj)| USp2g(C) j=1   2 g g 2g Z π Z π 1 1  X 2 Y  = g ··· g cosec θj g!π 2 |cos θk − cos θj| 0 0 j=1 k=1 k6=j g Y 2 Y 2 × (cos θn − cos θm) sin θ` dθ1 ··· dθg, 1≤m

g(g−1) Z π Z π g−1 2 2 Y 1 g g ··· cosec θg g!π |cos θk − cos θg| 0 0 k=1 g Y 2 Y 2 × (cos θn − cos θm) sin θ` dθ1 ··· dθg 1≤m

We now study the limit as n tends to infinity of the average

# {C ∈ H n : C satisfies P´olya’s Conjecture} 2g+1,q , #H2g+1,qn 7.1 Averages over Families of Curves 83

which we write as # {C ∈ H n ∩ P´olya} 2g+1,q . #H2g+1,qn Proposition 7.6 (cf. Proposition 4.11). We have that

# {C ∈ H2g+1,qn ∩ P´olya} lim = µHaar (ψ+(U) ≤ 0) . n→∞ #H2g+1,qn Proof. For any ε > 0, we may write

# {C ∈ H2g+1,qn ∩ P´olya} = A1 + A2 + A3 + A4 + A5 + A6 + A7, where

A1 = # {C ∈ H2g+1,qn : ψ+ (ϑ (C/Fqn )) ≤ 0} ,

A2 = −# {C ∈ H2g+1,qn \ LI : ψ+ (ϑ (C/Fqn )) ≤ 0} ,

A3 = # {C ∈ H2g+1,qn ∩ P´olya \ LI} ,  + A4 = # C ∈ H2g+1,qn ∩ LI : B (C/Fqn ) ≤ 0, 0 < ψ+ (ϑ (C/Fqn )) ≤ ε ,  + A5 = −# C ∈ H2g+1,qn ∩ LI : B (C/Fqn ) > 0, −ε ≤ ψ+ (ϑ (C/Fqn )) ≤ 0 ,  + A6 = # C ∈ H2g+1,qn ∩ LI : B (C/Fqn ) ≤ 0, ψ+ (ϑ (C/Fqn )) > ε ,  + A7 = −# C ∈ H2g+1,qn ∩ LI : B (C/Fqn ) > 0, ψ+ (ϑ (C/Fqn )) < −ε . Then we have that

A1 lim = µHaar(ψ+(U) ≤ 0), n→∞ #H2g+1,qn A A lim 2 = lim 3 = 0. n→∞ #H2g+1,qn n→∞ #H2g+1,qn Furthermore,

|A | + |A | # {C ∈ H n : −ε ≤ ψ (ϑ (C/ n )) ≤ ε} lim sup 4 5 ≤ lim 2g+1,q + Fq n→∞ #H2g+1,qn n→∞ #H2g+1,qn

= µHaar(−ε ≤ ψ+(U) ≤ ε). Finally, Lemma 7.1 implies the existence of a constant c(g) > 0 such that

 n/2 |A6| + |A7| ≤ # C ∈ H2g+1,qn : −ψ− (ϑ (C/Fqn )) ≥ εc(g)q .

0 As ψ− is integrable by Lemma 7.3, for any ε > 0 there exists some T0 > 0 such 0 that µHaar (−ψ−(U) ≥ T ) ≤ ε for all T > T0, and hence

|A | + |A | # {C ∈ H n : −ψ (ϑ (C/ n )) ≥ T } lim sup 6 7 ≤ lim 2g+1,q − Fq n→∞ #H2g+1,qn n→∞ #H2g+1,qn

= µHaar (−ψ−(U) ≥ T ) ≤ ε0. 84 Global P´olya Conjectures

As ε0 > 0 was arbitrary, A A lim 6 = lim 7 = 0. n→∞ #H2g+1,qn n→∞ #H2g+1,qn Thus for any ε > 0,

# {C ∈ H2g+1,qn ∩ P´olya} lim − µHaar (ψ+(U) ≤ 0) n→∞ #H2g+1,qn

≤ µHaar (−ε ≤ ψ+(U) ≤ ε) , which yields the result because

lim µHaar (−ε ≤ ψ+(U) ≤ ε) = µHaar (ψ+(U) = 0) = 0. ε→0

Proof of Theorem 5.9. We must show that µHaar (ψ+(U) ≤ 0) = 0. By making the change of variables cos θj 7→ xj, this is equivalent to showing that the set of g (x1, . . . , xg) ∈ [−1, 1] for which the function

ψe+(x1, . . . , xg) g g g g 1 Y 1 1 Y 1 1 X 1 Y 1 = − g+1 + g+1 + g 2 2 (1 − xj) 2 (1 + xj) 2 1 − x |xk − xj| j=1 j=1 j=1 j k=1 k6=j is nonpositive has measure zero with respect to the measure

g2 g 2 Y 2 Y q dµ (x , . . . , x ) = (x − x ) 1 − x2 dx ··· dx eUSp 1 g g!πg k j ` 1 g 1≤j

g on [−1, 1] . Now we may write ψe+ = f/h, with g g Y Y Y Y f(x1, . . . , xg) = − (1 − xj) |x` − xk| + (1 + xj) |x` − xk| j=1 1≤k<`≤g j=1 1≤k<`≤g g g X Y 2 Y + 2 (1 − xk) |xm − x`|, j=1 k=1 1≤`

g g Note that h is positive on [−1, 1] outside the µeUSp-measure zero subset of [−1, 1] where either xj = ±1 for some 1 ≤ j ≤ g or x` = xk for some 1 ≤ k < ` ≤ g. So it suffices to show that the set

g {(x1, . . . , xg) ∈ [−1, 1] : f(x1, . . . , xg) ≤ 0} 7.2 Variants of P´olya’s Conjecture 85

has µeUSp-measure zero. As f(x1, . . . , xg) is invariant under a permutation σ of {1, . . . , g}, we will be done if we can show that for each such permutation σ, the function f(x1, . . . , xg) is always positive on the set

 g (x1, . . . , xg) ∈ [−1, 1] : −1 < xσ(1) < . . . < xσ(g) < 1 .

For g = 1, this is elementary, as

f(x1) = 2 (1 − x1) , which is always positive for −1 < x1 < 1. In fact, one can show that 1 lim ψe+(x1) = , x1→1 4 and that this is the global minimum of ψe+(x1). For g = 2,  2
4 1 − x2 when −1 < x1 < x2 < 1, f(x1, x2) = 2
4 1 − x1 when −1 < x2 < x1 < 1, and in particular is always positive when x1, x2 6= ±1. Furthermore, it can be shown that  x1  27 lim ψe+ x1, − = , x1→1 3 64 and that this is the global minimum of ψe+(x1, x2).

Already when g = 3, the calculations become extremely complicated. How- ever, numerical calculations suggest that the global minimum of ψ+ is approxi- mately 0.530915.

7.2 Variants of P´olya’s Conjecture

Just as we discussed the α- and β-variants of the Mertens conjecture, we may do the same for P´olya’s conjecture. Here the properties of the weighted sum

X λ(n) L (x) = α nα n≤x for α ∈ R have recently been studied by Mossinghoff and Trudgian [20]; they ask whether for fixed α such sums are of constant sign for sufficiently large x. 86 Global P´olya Conjectures

For α > 1, the inequality Lα(x) > 0 will always hold for sufficiently large x, and indeed, Lα(x) converges to the absolutely convergent infinite series

∞ X λ(n) ζ(2α) = , nα ζ(α) n=1 which is strictly positive. Under the assumption of the Riemann hypothesis, this infinite series is conditionally convergent for 1/2 < α < 1, and as ζ(2α)/ζ(α) is negative in this range, we would expect the inequality Lα(x) < 0 to hold for all sufficiently large x. For 0 ≤ α < 1/2 and α = 1, the eventual constancy of sign of Lα(x) implies the Riemann hypothesis and the simplicity of the zeroes of the Riemann zeta function. However, Mossinghoff and Trudgian modify a result of Ingham [12] to show that the Linear Independence hypothesis for the Riemann zeta function implies that Lα(x) changes sign infinitely often for these values of

α; consequently, we would expect Lα(x) to change sign infinitely often for α in this range. Finally, for α = 1/2, Mossinghoff and Trudgian mimic the proof of Proposition 5.1 in order to show that

0 iγ log x γ0 ζ (1/2) X ζ(2ρ) x L (x) = + − + + R (x, T ) 1/2 2ζ(1/2) ζ(1/2) 2ζ(1/2)2 ζ0(ρ) iγ 1/2 v |γ|

Conjecture 7.7 (The α = 1/2 Conjecture [20, Problem 3]). For all x ≥ 17,

X λ(n) L (x) = √ ≤ 0. 1/2 n n≤x Here we study the function field analogue of this problem, namely for which α ∈ R the weighted sum

X−1 X 1 X L (X) = λ (D) C/Fq,α qα(N+1) C/Fq N=0 deg(D)=N 7.2 Variants of P´olya’s Conjecture 87

is of constant sign. For α > 1/2, the weighted sum LC/Fq,α(X) converges to the infinite series

∞ −2α 1 X 1 X ZC/Fq (q ) α αN µC/Fq (D) = α −α . q q q ZC/ (q ) N=0 deg(D)=N Fq

Thus for α > 1, LC/Fq,α(X) is eventually positive, while LC/Fq,α(X) is eventually negative in the range 1/2 < α < 1; however, LC/Fq,1(X) converges to zero, as −1 ZC/Fq (u) has a pole at u = q , so further analysis is necessary to determine sign changes for this particular weighted sum. For α < 1/2, we divide (6.3) by qα(N+1) and sum over all 0 ≤ N ≤ X − 1, showing that when ZC/Fq (u) has only simple zeroes,

LC/Fq,α(X) q(1/2−α)X √ √ −g √ √ −g 1 q q − 1 q hC/Fq X 1 q q + 1 q hC/Fq = − √ α √ −1/2 − (−1) √ α √ −1/2 2 q − q q + 1 PC/Fq (q ) 2 q + q q − 1 PC/Fq (−q ) −2   X ZC/ q (γ ) γ 1 − F + O . Z 0 (γ−1) γ − qα q,g q(1/2−α)X γ C/Fq

The proof of Lemma 7.1 can then be modified to show that as q tends to infinity, the quantity

+ LC/Fq,α(X) Bα (C/Fq) = lim sup (1/2−α)X X→∞ q satisfies the asymptotic

 1  B+(C/ ) = ϕ (ϑ(C/ )) + O − ϕ (ϑ(C/ )) α Fq + Fq g,α q1/2−α − Fq when C satisfies LI, and consequently the proof of Theorem 5.9 is still true with

P´olya’s conjecture for the function field C/Fq replaced by the conjecture that for fixed α < 1/2,

lim sup LC/Fq,α(X) ≤ 0. X→∞ It remains to study the function field version of the α = 1/2 conjecture. By dividing (6.3) by q(N+1)/2 and summing over all 0 ≤ N ≤ X − 1, we are able to determine the following expression for LC/Fq,1/2(X) when ZC/Fq (u) has only simple zeroes.

Proposition 7.8. Let C be a nonsingular projective curve over Fq of genus g ≥ 0, −1 and suppose that all of the zeroes γ of ZC/Fq (u) are simple. Then for each 88 Global P´olya Conjectures

X ≥ 1,

√ −g X √ −g 1 q − 1 q hC/Fq (−1) − 1 q + 1 q hC/Fq LC/Fq,1/2(X) = − √ −1/2 X − √ −1/2 2 q + 1 PC/Fq (q ) 4 q − 1 PC/Fq (−q ) −2 X ZC/ q (γ ) sin (Xθ(γ)/2) − F ei(X+1)θ(γ)/2 Z 0 (γ−1) sin (θ(γ)/2) γ C/Fq X −X/2 2 (−1) q − 1 q + 1 hC/ + √ Fq + R (q, g, T ), (7.1) q + 1 q − 1 h X,1/2 C/Fq2 where the sum is over the inverse zeroes of ZC/Fq (u), T > 0 is sufficiently small, and the error term RX,1/2(q, g, T ) is constant for X ≥ max{2g − 3, 1}.

The notable difference here to the number field case is that there are only

finitely many zeroes of ZC/Fq (u), and hence the sum over the inverse zeroes is bounded. Consequently, we have that

√ −g 1 q − 1 q hC/Fq LC/Fq,1/2(X) = − √ −1/2 X + Oq,g(1) 2 q + 1 PC/Fq (q ) as X tends to infinity, which resolves the function field analogue of the α = 1/2 conjecture.

Theorem 7.9 (The α = 1/2 Conjecture in Function Fields). Let C be a nonsin- gular projective curve over Fq of genus g ≥ 0, and suppose that all of the zeroes −1 γ of ZC/Fq (u) are simple. Then for all sufficiently large X, the inequality

X−1 X 1 X L (X) = λ (D) < 0 (7.2) C/Fq,1/2 q(N+1)/2 C/Fq N=0 deg(D)=N holds. Appendix A

Proof of the Kronecker–Weyl Theorem

In this appendix, we prove the following lemma.

Lemma 2.7 (Kronecker–Weyl Theorem). Let t1, . . . , tg be real numbers, and let H be the topological closure in Tg of the subgroup

 2πiXt1 2πiXtg
g He = e , . . . , e ∈ T : X ∈ Z .

g Then H is a closed subgroup of T . In particular, when the collection 1, t1, . . . , tg is linearly independent over the rational numbers, H is precisely Tg. Furthermore, g for arbitrary t1, . . . , tg and for any continuous function h : T → C, we have that 1 Y Z X 2πiXt1 2πiXtg
lim h e , . . . , e = h(z) dµH (z), Y →∞ Y X=1 H where µH is the normalised Haar measure on H.

The proof of this result makes use of several notable properties of Tg; namely that it is an abelian group that is also compact as a topological space. It is no surprise then that the method of proof uses abstract harmonic analysis. We must therefore first recall some definitions and results from this field.

Lemma A.1 ([7, Lemma 1.1.3]). Let He be a subgroup of a locally compact abelian group G. Then its topological closure H in G is also a subgroup of G.

Corollary A.2. Let t1, . . . , tg be arbitrary real numbers, and let H be the topo- logical closure in Tg of

 2πiXt1 2πiXtg
g+1 He = e , . . . , e ∈ T : X ∈ Z .

Then H is a closed subgroup of Tg.

89 90 Proof of the Kronecker–Weyl Theorem

Proof. Indeed, He is the image of the group homomorphism φ : Z → Tg given 2πiXt 2πiXt
g by φ(X) = e 1 , . . . , e g , and so He is a subgroup of T . The result then follows by Lemma A.1.

Definition A.3. Let G be a locally compact abelian group. A character on G is a continuous group homomorphism χ : G → T, where T = {z ∈ C : |z| = 1} is the circle group. The set of all characters on G is called the dual group of G and is denoted Gb. Proposition A.4 ([7, Theorem 3.2.1]). Let G be a locally compact abelian group. Then the dual group Gb of G is also a locally compact abelian group. Theorem A.5 (Pontryagin Duality [7, Theorem 3.5.5]). Let G be a locally com- pact abelian group. Then the dual group Gb of Gb is canonically isomorphic to G via the isomorphism x 7→ δx, where δx(χ) = χ(x) for each χ ∈ Gb.

The importance of showing earlier that H is a closed subgroup of Tg becomes evident through certain results involving the annihilator of H.

Definition A.6. Let H be a closed subgroup of a locally compact abelian group ⊥ G. The annihilator H of H is the set of all characters χ ∈ Gb satisfying χ|H = 1. Proposition A.7 ([7, Lemma 3.6.1]). Let G be a locally compact abelian group, and H a closed subgroup of G. Then H⊥ is isomorphic to G/H[ via the isomor- ⊥ phism χ 7→ χe, where χe(xH) = χ(x) for all xH ∈ G/H, and G/Hb is isomorphic ⊥ to Hb via the isomorphism χH 7→ χ|H .

Finally, we must determine exactly the characters of Tg and its dual group.

Lemma A.8. Let Tg be the g-torus. Then a character χ : Tg → T is of the form

k1 kg χ(z1, . . . , zg) = z1 ··· zg

g g for some (k1, . . . , kg) ∈ Z . Conversely, for any (k1, . . . , kg) ∈ Z , χ is a character of Tg. In particular, the dual group of Tg is isomorphic to Zg.

Of course, an analogous result holds for Zg.

Corollary A.9. A character χ : Zg → T is of the form

k1 kg χ(k1, . . . , kg) = z1 ··· zg

g g for some (z1, . . . , zg) ∈ T . Conversely, for any (z1, . . . , zg) ∈ T , χ is a character of Zg. In particular, the dual group of Zg is isomorphic to Tg. 91

We now have the framework necessary to determine H⊥ for H the topological g  2πiXt 2πiXt
g closure in T of the set He = e 1 , . . . , e g ∈ T : X ∈ Z .

Lemma A.10. Let t1, . . . , tg be arbitrary real numbers, and let H be the topo-  2πiXt 2πiXt
g g ⊥ logical closure of He = e 1 , . . . , e g ∈ T : X ∈ Z in T . Then H g is isomorphic to {k ∈ Z : t1k1 + ··· + tgkg ∈ Z}. In particular, if the collection g 1, t1, . . . , kg is linearly independent over the rational numbers, then H = T .

⊥ k1 kg Proof. Each character χ ∈ H is of the form χ(z1, . . . , zg) = z1 ··· zg for some g (k1, . . . , kg) ∈ Z with the property that for all X ∈ Z,

1 = χ e2πiXt1 , . . . , e2πiXtg
= e2πi(t1k1+···+tgkg)X , and hence t1k1 + ··· + tnkn ∈ Z. Conversely, if t1k1 + ··· + tnkn ∈ Z, then the k1 kg homomorphism χ(z1, . . . , zg) = z1 ··· zg satisfies χ|H = 1. Now the set g {k ∈ Z : t1k1 + ··· + tgkg ∈ Z} is isomorphic to

 g+1 k ∈ Z : k0 + t1k1 + ··· + tgkg = 0 , and if 1, t1, . . . , kg forms a linearly independent collection over the rational num- bers, then this is equal to the set {k = 0}. Thus H⊥ ∼= {0}, and hence H = Tg.

Next, we show that for any trigonometric polynomial h on Tg,

1 Y Z X 2πiXt1 2πiXtg
lim h e , . . . , e = h(z) dµH (z). Y →∞ Y X=1 H

The proof again makes use of the properties of the annihilator of H, this time via the Poisson summation formula.

Definition A.11. Let G be a locally compact abelian group, and let h : G → C be a continuous compactly supported function. The Fourier transform of h with respect to a Haar measure µG on G is the function bh on Gb given by Z bh(χ) = h(x)χ(x) dµG(x). G 92 Proof of the Kronecker–Weyl Theorem

Proposition A.12 (Poisson Summation Formula [7, Theorem 3.6.3]). Let H be a closed subgroup of a locally compact abelian group G, and let h : G → C be a continuous compactly supported function. Then we have that Z Z h(z) dµH (z) = bh(χ) dµH⊥ (χ), H H⊥ ⊥ where µH is a Haar measure on H and µH⊥ is the induced Haar measure on H .

g Lemma A.13. Let t1, . . . , tg be arbitrary real numbers, and let h : T → C be a trigonometric polynomial; that is, a function of the form

X k1 kg h(z) = ckz1 ··· zg k∈Zg g for z = (z1, . . . , zg) ∈ T , where all but finitely many of the coefficients ck ∈ C are zero. Then we have that 1 Y Z X 2πiXt1 2πiXtg
lim h e , . . . , e = h(z) dµH (z), Y →∞ Y X=1 H where µH is the normalised Haar measure on H.

Proof. Let χ : Tg → T be a character corresponding to ek ∈ Zg. Then Z Z Z X k1 kg kf1 kfg bh(χ) = h(z)χ(z) dz = ··· ckz1 ··· zg z1 ··· zg dz1 ··· dzg. g T T T k∈Zg We may interchange the order of summation and integration as there are only finitely many nonzero members in this sum. Thus g Z X Y kj −kej bh(χ) = ck zj dzj k∈Zg j=1 T g Z 1 X Y 2πi(kj −kej )θ = ck e dθ 0 k∈Zg j=1 g  X Y 1 if kj = kej, = ck k∈Zg j=1 0 otherwise, = c . ek ⊥ g Recalling that H is isomorphic to {k ∈ Z : t1k1 + ··· + tgkg ∈ Z}, so that the ⊥ Haar measure µH⊥ on H is simply the counting measure, we therefore obtain by the Poisson summation formula that Z Z X h(z) dµH (z) = bh(χ) dµH⊥ (χ) = ck. H H⊥ g k∈Z t1k1+···+tgkg∈Z 93

On the other hand,

Y X h e2πiXt1 , . . . , e2πiXtg
X=1 Y X X 2πi(t1k1+···+tgkg)X = ck e k∈Zg X=1 2πi(t1k1+···+tgkg)Y
X X ck e − 1 = ckY + . −2πi(t1k1+···+tgkg) g g 1 − e k∈Z k∈Z t1k1+···+tgkg∈Z t1k1+···+tgkg∈/Z Thus

1 Y Z X 2πiXt1 2πiXtg
X lim h e , . . . , e = ck = h(z) dµH (z). Y →∞ Y g H X=1 k∈Z t1k1+···+tgkg∈Z From this, we may easily obtain the result in the general case where h is merely a continuous function. Indeed, this follows simply from the density of the trigonometric polynomials in the space of continuous complex-valued functions on Tg with regards to the supremum norm, that is to say, the Stone–Weierstrass theorem.

Lemma A.14. For any continuous function h : Tg → C, we have that

1 Y Z X 2πiXt1 2πiXtg
lim h e , . . . , e = h(z) dµH (z). Y →∞ Y X=1 H Proof. Given a continuous function h : Tn → C and a fixed ε > 0, the Stone– Weierstrass theorem shows the existence of a trigonometric polynomial

X k1 kg eh(z) = ckz1 ··· zg , k∈Zg where all but finitely many of the coefficients ck ∈ C are zero, such that

max h(z) − eh(z) < ε/2. z∈Tg Then

1 Y ε X 2πiXt1 2πiXtg
2πiXt1 2πiXtg
lim h e , . . . , e − eh e , . . . , e < , Y →∞ Y 2 X=1 and similarly Z ε

h(z) − eh(z) dµH (z) < H 2 94 Proof of the Kronecker–Weyl Theorem

as µH (H) = 1, and consequently

1 Y Z X 2πiXt1 2πiXtg
lim h e , . . . , e − h(z) dµH (z) < ε. Y →∞ Y X=1 H As ε > 0 was arbitrary, we obtain the result.

This completes the proof of the Kronecker–Weyl theorem. Bibliography

[1] P. T. Bateman, J. W. Brown, R. S. Hall, K. E. Kloss, and Rosemarie M. Stemmler, “Linear Relations Connecting the Imaginary Parts of the Zeros of the Zeta Function”, in Computers in Number Theory, editors A. O. L. Atkin and B. J. Birch, Academic Press, London, 1971, 11–19.

[2] Patrick Billingsley, Convergence of Probability Measures, 2nd Edition, John Wiley and Sons, New York, 1999.

[3] Peter Borwein, Ron Ferguson, and Michael J. Mossinghoff, “Sign Changes in Sums of the Liouville Function”, Mathematics of Computation 77 (2008), no. 263, 1681–1694.

[4] Byungchul Cha, “Chebyshev’s Bias in Function Fields”, Compositio Mathe- matica 144 (2008), no. 6, 1351–1374.

[5] Byungchul Cha, “The Summatory Function of the M¨obiusFunction in Func- tion Fields”, submitted for publication, arXiv:math.NT/1008.4711v2 (14 November 2011), 16 pages.

[6] Nick Chavdarov, “The General Irreducibility of the Numerator of the Zeta Function in a Family of Curves with Large Monodromy”, Duke Mathematical Journal 87 (1997), no. 1, 151–180.

[7] Anton Deitmar and Siegfried Echterhoff, Principles of Harmonic Analysis, Universitext, Springer, New York, 2009.

[8] A. Y. Fawaz, “The Explicit Formula for L0(x)”, Proceedings of the London Mathematical Society 1 (1951), no. 3, 86–103.

[9] C. B. Haselgrove, “A Disproof of a Conjecture of P´olya”, Mathematika 5 (1958), 141–145.

95 96 Bibliography

[10] C. P. Hughes, J. P. Keating, and Neil O’Connell, “Random Matrix Theory and the Derivative of the Riemann Zeta-Function”, Proceedings of the Royal Society of London Serial A 456 (2000), 2611–2627.

[11] Peter Humphries, “The Distribution of Weighted Sums of the Liouville Function and P´olya’s Conjecture”, to appear in Journal of Number Theory, arXiv:math.NT/1108.1524 (7 August 2011), 32 pages.

[12] A. E. Ingham, “On Two Conjectures in the Theory of Numbers”, American Journal of Mathematics 64 (1942), 313–319.

[13] Nicholas M. Katz and Peter Sarnak, Random Matrices, Frobenius Eigenval- ues, and Monodromy, American Mathematical Society Colloquium Publica- tions 45, American Mathematical Society, Providence, 1999.

[14] J. P. Keating and N. C. Snaith, “Random Matrix Theory and L-functions at s = 1/2”, Communications in Mathematical Physics 214 (2000), no. 1, 91–110.

[15] Tadej Kotnik and Herman te Riele, “The Mertens Conjecture Revisited”, in Algorithmic Number Theory; 7th International Symposium, ANTS-VII; Berlin, Germany, July 2006; Proceedings, editors Florian Hess, Sebas- tian Pauli, and Michael Pohst, Lecture Notes in Computer Science 4076, Springer, Berlin, 2006, 156–167.

[16] Emmanuel Kowalski, “The Large Sieve, Monodromy, and Zeta Functions of Algebraic Curves, 2: Independence of the Zeros”, International Mathematics Research Papers (2008), Article ID rnn091, 57 pages.

[17] F. Mertens, “Uber¨ eine zahlentheoretische Funktion”, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Mathematisch-Naturwissen- schaftliche Klasse, Abteilung 2a 106 (1897), 761–830.

[18] Hugh L. Montgomery, “The Zeta Function and Prime Numbers”, in Proceed- ings of the Queen’s Number Theory Conference, 1979, editor P. Ribenboim, Queen’s Papers in Pure and Applied Mathematics 54, Queen’s University, Kingston, Ontario, 1980, 1–31.

[19] Hugh L. Montgomery and Robert C. Vaughan, Multiplicative Number The- ory: I. Classical Theory, Cambridge Studies in Advanced Mathematics 97, Cambridge University Press, Cambridge, 2007. Bibliography 97

[20] Michael J. Mossinghoff and Timothy S. Trudgian, “Between the Problems of P´olya and Tur´an”,to appear in Journal of the Australian Mathematical Soci- ety, http://www.davidson.edu/math/mossinghoff/BetweenPolyaTuran_ MT.pdf (28 September 2011), 13 pages.

[21] Nathan Ng, “The Distribution of the Summatory Function of the M¨obius Function”, Proceedings of the London Mathematical Society 89 (2004), 361– 389.

[22] A. M. Odlyzko and H. J. J. te Riele, “Disproof of the Mertens Conjecture”, Journal f¨urdie Reine und Angewandte Mathematik 357 (1985), 138–160.

[23] Georg P´olya, “Verschiedene Bemerkungen zur Zahlentheorie”, Jahresbericht der Deutschen Mathematiker-Vereinigung 28 (1919), 31–40.

[24] Michael Rosen, Number Theory in Function Fields, Graduate Texts in Math- ematics 210, Springer, New York, 2002.

[25] Michael Rubinstein and Peter Sarnak, “Chebyshev’s Bias”, Experimental Mathematics 3 (1994), no. 3, 173–197.

[26] R. D. von Sterneck, “Neue empirische Daten ¨uber die zahlentheoretischen Funktion σ(n)”, in Proceedings of the Fifth International Congress of Math- ematics, Cambridge, 22–28 August 1912, Volume 1, editors E. W. Hobson and A. E. H. Love, Cambridge, 1913, 341–343.

[27] T. J. Stieltjes, Lettre `aHermite de 11 juillet 1885, Lettre #79, in Correspon- dance d’Hermite et de Stieltjes, Tome 1, editors B. Baillaud and H. Bourget, Paris, Gauthier–Villars, 1905, 160–164.

[28] M. Tanaka, “A Numerical Investigation on Cumulative Sum of the Liouville Function”, Tokyo Journal of Mathematics 3 (1980), 187–189.

[29] William C. Waterhouse, “Abelian Varieties over Finite Fields”, Annales Sci- entifiques de l’Ecole´ Normale Sup´erieure 4 (1969), t. 2, 521–560.